A BIJECTION FOR COVERED MAPS, OR A SHORTCUT
BETWEEN HARER-ZAGIER’S AND JACKSON’S FORMULAS.
OLIVIER BERNARDI
1
AND GUILLAUME CHAPUY
2
Abstract. We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g) with a marked
unicellular spanning submap (which can have any genus in {0, 1, . . . , g}). Our
main result is a bijection between covered maps with n edges and genus g and
pairs made of a plane tree with n edges and a unicellular bipartite map of genus
g with n + 1 edges. In the planar case, covered maps are maps with a marked
spanning tree and our bijection specializes into a construction obtained by the
first author in [4].
Covered maps can also be seen as shuffles of two unicellular maps (one
representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of
unicellular maps, and pairs made of a plane tree and a unicellular bipartite
map. In terms of counting, this establishes the equivalence between a formula
due to Harer and Zagier for general unicellular maps, and a formula due to
Jackson for bipartite unicellular maps.
We also show that the bijection of Bouttier, Di Francesco and Guitter [8]
(which generalizes a previous bijection by Schaeffer [33]) between bipartite
maps and so-called well-labelled mobiles can be obtained as a special case of
our bijection.
1. Introduction.
We consider maps on orientable surfaces of arbitrary genus. A map is called
unicellular if it has a single face. For instance, the unicellular maps of genus 0
are the plane trees. A covered map is a map together with a marked unicellular
spanning submap. A map of genus g can have spanning submaps of any genus in
{0 . . . , g}. In particular, a tree-rooted map (map with a marked spanning tree) is
a covered map since the spanning trees are the unicellular spanning submaps of
genus 0. A covered map of genus 2 with a unicellular spanning submap of genus 1
is represented in Figure 1(a).
Our main result is a bijection (denoted by Ψ) between covered maps of genus g
with n edges, and pairs made of a plane tree with n edges and a bipartite unicellular
map of genus g with n + 1 edges. If the covered map has v vertices and f faces,
then the bipartite unicellular map has v white vertices and f black vertices. The
Date: March 14, 2011.
1 Massachussets Institute of Technology, Department of Mathematics, Cambridge MA 02139,
USA. Partial support from the French grant ANR A3 and the European grant ERC Explore maps.
bernardi@math.mit.edu.
2 Department of Mathematics, Simon Fraser University, Burnaby BC V5A 1S6, Canada. Partial
support from the a CNRS/PIMS postdoctoral fellowship, and the European grant ERC Explore
maps. gchapuy@sfu.ca .
1
2
O. BERNARDI AND G. CHAPUY
bijection Ψ is represented in Figure 1. In the planar case g = 0, the bijection
Ψ coincides with a construction by the first author [4] between planar tree-rooted
maps with n edges and pairs of plane trees with n and n + 1 edges respectively1.
As explained below, the bijection Ψ also extends several other bijections.
Ψ
(a)
(b)
Figure 1. (a) A covered map of genus 2 (the unicellular submap
of genus 1 is drawn in thick lines). (b) The image of the covered
map by the bijection Ψ is made of a tree and a bipartite unicellular
map of genus 2 (the mobile associated to the covered map).
Before discussing the bijection Ψ further, let us give another equivalent way of
defining covered maps. We start with the planar case which is simpler. Let M be
a planar map and let M ∗ be its dual. Then, for any spanning tree T of M , the
dual of the edges not in T form a spanning tree of M ∗ . In other words, the dual
of a planar tree-rooted map is a planar tree-rooted map. Pushing this observation
further, Mullin showed in [32] that a tree-rooted map can be encoded by a shuffle
of two trees (one representing the spanning tree on primal map M , the other representing the spanning tree on the dual map M ∗ ), or more precisely as a shuffle of two
parenthesis systems encoding these trees. Covered maps enjoy a similar property:
the dual of a covered map is a covered map. Using this observation, it is not hard
to see that covered maps can be encoded by shuffles of two unicellular maps (see
Section 3).
We emphasize that our bijection Ψ is not the encoding of covered maps as shuffles
of two unicellular maps: the image by Ψ is a pair of unicellular maps of a fixed
size, and not a shuffle. As a matter of fact, comparing the enumerative formulas
given by the bijection Ψ with the formulas given by the shuffle approach yields the
equivalence between formulas by Harer and Zagier [20], and by Jackson [21]. As a
warm up, let us consider the total number C(n) of covered maps with n edges (and
arbitrary genus). The bijection Ψ implies
(1)
C(n) = Cat(n)B(n + 1) =
(2n)!
,
n!
(2n)!
is the Catalan number counting rooted plane trees with n
where Cat(n) = n!(n+1)!
edges, and B(n + 1) = (n + 1)! is the total number of bipartite unicellular maps
1In [4], the tree with n + 1 edges was actually described as a non-crossing partition.
A BIJECTION FOR COVERED MAPS
3
with n + 1 edges. On the other hand, the shuffle approach implies
n X
2n
C(n) =
(2)
A(k)A(n − k),
2k
k=0
(2k)!
2k k!
is the total number of unicellular maps with k edges (and the
where A(k) =
term 2n
accounts
for the shuffling). Here, the relation between (1) and (2) is
2k
merely an application of the Chu-Vandermonde identity, but things get more interesting when one considers refinements of these equations. First, by the bijection Ψ,
the number of covered maps of genus g with n edges is
(3)
Cg (n) = Cat(n)Bg (n + 1),
where Bg (n + 1) is the number of bipartite unicellular maps of genus g with n + 1
edges. A further refinement can be obtained by taking into account the numbers
p and q of vertices and faces of the covered map. We show in Section 5, that
comparing the formula given by the bijection Ψ with the formula given by the
shuffle approach yields
n
X
n!(n + 1)!
(4)
Ap (k)Aq (n − k),
B p,q (n + 1) =
(2k)!(2n − 2k)!
k=0
p,q
where B (n + 1) is the number of unicellular bipartite maps with n + 1 edges, p
white vertices and q black vertices, and Ap (k) is the number of unicellular maps with
n edges and p vertices. Equation (4) actually establishes the equivalence between
the formula of Harer and Zagier [20] for unicellular maps:
X
(2n)! X i−1
n
y
Ap (n)y p = n
2
(5)
,
2 n!
i−1
i
p≥1
i≥1
and the formula of Jackson [21] for bipartite unicellular maps:
X
X n
y
z
p,q
p q
B (n + 1)y z = (n + 1)!
(6)
.
i − 1, j − 1
i
j
p,q≥1
i,j≥1
The original proof of (5) involves a matrix integral argument; combinatorial proofs
are given in [23, 18, 3, 11]. The original proof of (6) (as well as another related
formula by Adrianov [1]) is based on the representation theory of the symmetric
group; a combinatorial proof is given in [34].
Let us mention a few other enumerative corollaries of the bijection Ψ. First,
plugging the identity2 A1 (k) = A(k)/(k + 1) in the case q = 1 of Equation (4)
shows that the number B p (n) of bipartite unicellular maps with n edges and p
white vertices satisfies
n(n + 1) p,1
B (n).
B p (n + 1) =
2
This formula is originally due to Zagier [35], and has been given a bijective proof
(different from ours) by Feray and Vassilieva [17]. We now turn to formulas concerning the number Tg (n) of tree-rooted maps of genus g with n edges. In the
planar case, tree-rooted maps are the same as covered maps. Hence, (3) gives
(7)
T0 (n) = Cat(n)Cat(n + 1),
2This identity is originally due to Lehman and Walsh [27, Eq. (14)] and can easily be proved
bijectively by using the results from [11].
4
O. BERNARDI AND G. CHAPUY
This formula was originally proved by Mullin [32] (using the shuffle approach) who
asked for a direct bijective proof; this was the original motivation for the planar
specialization of Ψ described in [4] (and for a related recursive bijection due to Cori,
Dulucq and Viennot [15]). In the case of genus g = 1, a duality argument shows
that exactly half of the covered maps are tree-rooted maps, so that (3) gives
(8)
T1 (n) =
(2n)!(2n − 1)!
1
Cat(n)B1 (n + 1) =
,
2
12(n + 1)!n!(n − 1)!(n − 2)!
This formula was originally proved by Lehman and Walsh (using the shuffle approach), and had no direct bijective proof so far.
We now explain the relation between the bijection Ψ and some known bijections.
In [5, 6], two “master bijections” for planar maps are defined, and then specialized
in various ways so as to unify and extend several known bijections (roughly speaking, by specializing the master bijections appropriately, one can obtain a bijection
for any class of planar maps defined by a girth condition and a face-degree condition). The master bijections are in fact “variants” of the planar version of the
bijection Ψ. Here we shall prove that the bijection Ψ generalizes the bijection obtained for bipartite planar maps by Bouttier, Di Francesco and Guitter [8, Section
2] (however, we do not recover the most general version of their bijection [8, Section 3-4]), as well as its generalization to higher genus surfaces by Chapuy, Marcus
and Schaeffer [13, 12]. These bijections (which themselves generalize a previous
bijection by Schaeffer [33]) are of fundamental importance for studying the metric
properties of random maps [14, 7, 10, 9, 30] and for defining and analyzing their
continuous limit, the Brownian map [29, 24, 26, 25].
The paper is organized as follows. In Section 2, we recall some definitions about
maps. In Section 3, we show that covered maps can be encoded by shuffles of unicellular maps. In Section 4, we define the bijection Ψ between covered maps with
n edges and pairs made of a plane tree with n edges and a bipartite unicellular
map with n + 1 edges. In Section 5, we explore the enumerative implications of the
bijection Ψ. In Section 6, we prove the bijectivity of Ψ. In Section 7, we give three
equivalent ways of describing the image of Ψ (the pairs made of a plane tree and a
bipartite unicellular map) and use one of these descriptions in order to show that
the bijections for bipartite maps described in [8, 13] are specializations of Ψ. Lastly,
in Section 8, we study the properties of the bijection Ψ with respect to duality.
2. Definitions
Maps. Maps can either be defined topologically (as graphs embedded in surfaces)
or combinatorially (in terms of permutations). We shall prove our results using
the combinatorial definition, but resort to the topological interpretation in order to
convey intuitions.
We start with the topological definition of maps. Here, surfaces are 2-dimensional,
oriented, compact and without boundaries. A map is a connected graph embedded
in surface, considered up to orientation preserving homeomorphism. By embedded,
one means drawn on the surface in such a way that the edges do not intersect and
A BIJECTION FOR COVERED MAPS
5
the faces (connected components of the complement of the graph) are simply connected. Loops and multiple edges are allowed. The genus of the map is the genus
of the underlying surface and its size is its number of edges. A planar map is a
map of genus 0. A map is unicellular if it has a single face. For instance, the planar
unicellular maps are the plane trees. A map is bipartite if vertices can be colored in
black and white in such a way that every edge join a white vertex to a black vertex.
We denote by g(M ) the genus of a map M and by v(M ), f (M ), e(M ) respectively
its number of vertices, faces and edges. These quantities are related by the Euler
formula:
v(M ) − e(M ) + f (M ) = 2 − 2g(M ).
By removing the midpoint of an edge, one obtains two half-edges. Two consecutive half-edges around a vertex define a corner. A map is rooted if one half-edge
is distinguished as the root. The vertex incident to the root is called root-vertex.
In figures, the rooting will be indicated by an arrow pointing into the root-corner,
that is, the corner following the root in clockwise order around the root-vertex. For
instance, the root of the map in Figure 2 is the half-edge a1 .
σ
ā3
a3
a4
a5
ā5
ā2
a1
ā1
a2
ā4
φ
Figure 2. A unicellular map M = (H, σ, α) of genus 1. The
set of half-edges is H = {a1 , ā1 , . . . , a5 , ā5 }, the edge-permutation
is α = (a1 , ā1 ) · · · (a5 , ā5 ), the vertex-permutation is σ =
(a1 , ā2 , a3 )(ā1 , a2 , ā4 )(ā3 , a4 , a5 )(ā5 ), and the face-permutation is
φ = σα = (a1 , a2 , a3 , a4 , ā1 , ā2 , ā4 , a5 , ā5 , ā3 ).
Maps can also be defined in terms of permutations acting on half-edges. To
obtain this equivalence, observe first that the embedding of a graph in a surface
defines a cyclic order (the counterclockwise order) of the half-edges around each
vertex. This gives in fact a one-to-one correspondence between maps and connected graphs together with a cyclic order of the half-edges around each vertex
(see e.g. [31]). Equivalently, a map can be defined as a triple M = (H, σ, α) where
H is a finite set whose elements are called the half-edges, α is an involution of H
without fixed point, and σ is a permutation of H such that the group generated by
σ and α acts transitively on H (here we follow the notations in [16]). This must
be understood as follows: each cycle of σ describes the counterclockwise order of
the half-edges around one vertex of the map, and each cycle of α describes an edge,
that is, a pair of two half-edges; see Figure 2 for an example. The transitivity
assumption simply translates the fact that the graph is connected.
For a map M = (H, σ, α), the permutation σ is called vertex-permutation, the
permutation α is called edge-permutation and the permutation φ = σα is called
6
O. BERNARDI AND G. CHAPUY
face-permutation. The cycles of σ, α, φ are called vertices, edges and faces. Observe that the cycles of φ are indeed in bijection with the faces of the map in its
topological interpretation. Hence, the genus of M can be deduced from the number
of cycles of σ, α and φ by the Euler relation. We say that a half-edge is incident to
a vertex or a face if this edge belongs to the corresponding cycle. Again, a map is
rooted if one of the half-edges is distinguished as the root ; the incident vertex and
face are called root-vertex and root-face.
The correspondence between topological and combinatorial maps is one-to-one
if combinatorial maps are considered up to isomorphism (or, relabelling). That
is, two maps (H, σ, α) and (H ′ , σ ′ , α′ ) are considered the same if there exists a
bijection λ : H → H ′ such that σ ′ = λσλ−1 and α′ = λαλ−1 (for rooted maps,
we ask furthermore that λ(r) = r′ ). In this article all maps will be rooted, and
considered up to isomorphism.
We call pseudo map a triple M = (H, σ, α) such that α is a fixed-point free
involution, but where the transitivity assumption (i.e. connectivity assumption) is
not required. This can be seen as a union of maps and we still call φ = σα the
face-permutation, as its cycles are indeed in correspondence with the faces of the
union of maps. Lastly, we consider the case where the set of half-edges H is empty
as a special case of rooted unicellular map (corresponding to the planar map with
one vertex and no edge) called empty map.
Submaps, covered maps and motion functions. For a permutation π on a
set H, we call restriction of π to a set S ⊆ H and denote by π|S the permutation
of S whose cycles are obtained from the cycles of π by erasing the elements not in
S. Observe that (π −1 )|S = (π|S )−1 so that we shall not use parenthesis anymore
in these notations. It is sometime convenient to consider the restriction π|S as a
permutation on the whole set H acting as the identity on H \ S; we shall mention
this abuse of notations whenever necessary.
A spanned map is a map with a marked subset of edges. In terms of permutations, a spanned map is a pair (M, S), where S is a subset of half-edges stable
by the edge-permutation α. The submap defined by S, denoted M|S , is the pseudo
map (S, σ|S , α|S ), where σ is the vertex-permutation of M . We underline that the
face-permutation φS = σ|S α|S of the pseudo-map M|S is not equal to (σα)|S . Observe also that the genus of M|S can be less than the genus of M . For example,
Figure 1(a) represents a submap of genus 1 of a map of genus 2. A submap M|S
is connecting if it is a map containing every vertex of M , that is, S contains a
half-edge in every vertex of M (except if M has a single vertex, where we authorize
S to be empty) and σ|S , α|S act transitively on S. The submap represented in
Figure 3 (right) is a map but is not connecting. A covered map is a spanned map
such that the submap M|S is a connecting unicellular map. A tree-rooted map is a
spanned map such that the submap M|S is a spanning tree, that is, a connecting
plane tree.
The motion function of the spanned map (M, S) is the mapping θ defined on H
by θ(h) = φ(h) ≡ σα(h) if h is in S and θ(h) = σ(h) otherwise. Observe that the
stability of S by α implies that the motion function θ is a permutation of H since
A BIJECTION FOR COVERED MAPS
7
its inverse is given by θ−1 (h) = ασ −1 (h) if σ −1 (h) is in S and θ−1 (h) = σ −1 (h)
otherwise. Observe also that, given the map M , the set S can be recovered from
the motion function θ. Topologically, the motion function is the permutation describing the tour of the faces of the connected components of the submap M|S in
counterclockwise direction: we follow the border of the edges of the submap M|S
and cross the edges not in M|S . See Figure 3 for an example.
i
j
h
g
m
f
l
a
n
b
k
c
e
d
Figure 3. The submap M|S defined by S = {a, b, c, d, i, j, k, l},
and its motion function θ = (a, c, e, n, d, k, m, h, i)(b, j, l)(f, g). The
cycles of θ are represented in dashed lines.
Orientations. An orientation of a map M = (H, σ, α) is a partition H = I ⊎ O
such that the involution α maps the set I of ingoing half-edges to the set O of
outgoing half-edges. The pair (M, (I, O)) is an oriented map. A directed path is
a sequence h1 , h2 , . . . , hk of distinct ingoing half-edges such that hi , α(hi+1 ) are
incident to the same vertex (are in the same cycles of σ) for i = 1 . . . k − 1. A
directed cycle is a directed path h1 , . . . , hk such that hk and α(h1 ) are incident to
the same vertex. The half-edge hk is called the extremity of the directed path. An
orientation is root-connected if any ingoing half-edge h is the extremity of a directed
path h1 , . . . , hk = h such that α(h1 ) is incident to the root-vertex of M .
Duality The dual map of a map M = (H, σ, α) is the map M ∗ = (H, φ, α) where
φ = σα is the face-permutation of M . The root of the dual map M ∗ is equal to
the root of M . Observe that the genus of a map and of its dual are equal (by Euler
relation) and that M ∗∗ = M . Topologically, the dual map M ∗ is obtained by the
following two steps process: see Figure 4.
(1) In each face f of M , draw a vertex vf of M ∗ . For each edge e of M
separating faces f and f ′ (which can be equal), draw the dual edge e∗ of
M ∗ going from vf to vf ′ across e.
(2) Flip the drawing of M ∗ , that is, inverse the orientation of the surface.
We now define duals of spanned maps and oriented maps. Given a subset S ⊆ H,
we denote S̄ = H \ S. The dual of a spanned map (M, S) is the spanned map
(M ∗ , S̄); see Figure 4. We also say that M|S and M|∗S̄ are dual submaps. Observe
that the motion functions of a spanned map (M, S) and of its dual (M ∗ , S̄) are
equal. The dual of the oriented map (M, (I, O)) is (M ∗ , (I, O)). Graphically, this
orientation is obtained by applying the following rule at step 1: the dual-edge e∗
of an edge e ∈ M is oriented from the left of e to the right of e; see Figure 14.
Observe that duality is involutive on maps, spanned maps and oriented maps.
8
O. BERNARDI AND G. CHAPUY
a4
b̄3
b3
a3 b̄2
ā3
ā1
b̄3
ā4
a4
a1
ā3
b2
a2
ā4 b3 a3
ā2
b2
ā2
b̄1
b1
(a)
ā1
b̄2
a1
a2
b̄1
b1
(b)
(c)
Figure 4. (a) A spanned map (the submap is indicated by thick
lines). (b) Topological construction of the dual. (c) The dual
covered map.
3. Covered maps as shuffles of unicellular maps.
In this Section, we establish some preliminary results about covered maps. In
particular we prove that covered maps are stable by duality and explicit their
encoding as shuffles of two unicellular maps. Our first result should come as no
surprise: it simply states that a spanned map (M, S) is a covered map if and only if
turning around the submap M|S (that is following the border of its edges) starting
from the root allows one to visit every half-edge of M .
Proposition 3.1. Let (M, S) be a spanned map, and let σ, α and φ = σα be
the vertex-, edge-, and face-permutations of M . The motion function θ satisfies
θ|S = σ|S α|S and θ|S̄ = φ|S̄ α|S̄ . That is, the restriction θ|S is the face permutation
of the pseudo map M|S , while the restriction θ|S̄ is the face-permutation of the dual
pseudo map M|∗S̄ .
In particular a spanned map is a covered map if and only if its motion function
is a cyclic permutation.
Proposition 3.1 is a consequence of the following lemma:
Lemma 3.2. Let σ and φ be two permutations on the set {1, 2, . . . , n}, let S be a
subset of {1, 2, . . . , n}, and let θ be the mapping defined by:
θ(a) = σ(a) if a ∈ S and θ(a) = φ(a) if a 6∈ S.
Then θ is a permutation if and only if S is stable by φ−1 σ. Moreover in that case
we have:
θ|S = φ|S (φ−1 σ)|S .
Proof of Lemma 3.2. First, if S is stable by φ−1 σ then the inverse of θ is given by
θ−1 (a) = σ −1 (a) if σ −1 (S) ∈ S and θ−1 (a) = φ−1 (a) is σ −1 (S) 6∈ S (note that
σ −1 (S) 6∈ S is equivalent to φ−1 (a) 6∈ S since S is stable by φ−1 σ). Conversely, if
there exists s ∈ S such that φ−1 σ(s) 6∈ S, then θ(s) = σ(s) = θ(φ−1 σ(s)) and θ is
not a permutation. This proves the first claim.
For the second claim, let s ∈ S and r = θ|S (s). By definition of the restriction
we have r ∈ S, and there exist h1 , h2 , . . . , hk 6∈ S such that θ(s) = h1 , θ(hi ) = hi+1
for i < k, and θ(hk ) = r. Moreover by definition of θ we have θ(s) = σ(s) and
θ(hi ) = φ(hi ) for i ≤ k. Now, since S is stable by φ−1 σ, we have (φ−1 σ)|S (s) =
A BIJECTION FOR COVERED MAPS
9
φ−1 σ(s) = φ−1 (h1 ), which implies that φ|S (φ−1 σ)|S (s) = φ|S (h1 ) = r by definition
of the restriction.
Proof of Proposition 3.1. The fact θ|S = σ|S α|S comes from Lemma 3.2, and the
relation θ̄|S̄ = φ|S̄ α|S̄ follows from the preceding point by duality (since the motion
functions of a spanned map and its dual are equal).
Now let (M, S) be a covered map. Since M|S is connecting, each cycle of the
motion function θ contains an element of S. Hence, the number of cycles of θ and
θ|S is the same. Moreover, by Lemma 3.2, θ|S = σ|S α|S is the face-permutation of
M|S . Since M|S is unicellular, θ|S = σ|S α|S is cyclic and θ is also cyclic.
Conversely, suppose that the motion function θ is cyclic. In this case, the pseudo
map M|S has a face-permutation which is cyclic by Lemma 3.2. Hence it is a
unicellular map.
Proposition 3.1 immediately gives the following corollary concerning duality.
Corollary 3.3. If a spanned map (M, S) is a covered map, then the dual spanned
map (M ∗ , S̄) is also a covered map. Moreover the genus of M is the sum of the
genera of the unicellular maps M|S and M|∗S̄ :
g(M ) = g(M|S ) + g(M|∗S̄ ).
Corollary 3.3 is illustrated by Figure 4.
Proof. The fact that (M ∗ , S̄) is a covered map is an immediate consequence of
Proposition 3.1 since the motion function of a submap and of its dual are always
equal. The fact that the genus adds up is obtained by writing the Euler relation
for the maps M , M|S and M|∗S̄ .
Let (M, S) be a covered map. By Proposition 3.1, the restrictions θ|S and θ|S̄
of the motion function θ correspond respectively to the face-permutations of the
unicellular maps M|S and M|∗S̄ . This inclines to say, somewhat vaguely, that the
covered map (M, S) is a shuffle of the unicellular maps M|S and M|∗S̄ . Making this
statement precise requires introducing codes of unicellular maps and covered maps.
A unicellular code on the alphabet An = {a1 , ā1 , . . . , an , ān } is a word on An
such that every letter of An appears exactly once, and for all 1 ≤ i < j ≤ n, the
letter ai appears before āi and before aj . Let T = (H, σ, α) be a unicellular map
with n edges. By definition, the face-permutation φ = σα is cyclic. Hence, there
exists a unique way of relabelling the half-edges on the set An in such a way that
α(ai ) = āi for all i = 1 . . . n and φ = (w1 , w2 , . . . , w2n ), where w1 is the root and
w = w1 w2 · · · w2n is a unicellular code. We call w the code of the unicellular map T .
Topologically, the code of a unicellular map is obtained by turning around the
face of the map in counterclockwise direction starting from the root and writing
ai when we discover the ith edge and writing āi when we see this edge for the
second time. For instance, the code of the unicellular map in Figure 2 is w =
a1 a2 a3 a4 ā1 ā2 ā4 a5 ā5 ā3 . We also mention that the unicellular map T is a plane tree
if and only if its code w does not contain a subword of the form ai aj āi āj . In this
special case, replacing all the letters ai , i = 1 . . . n of the code w by the letter a and
all the letters āi , i = 1 . . . n by the letter ā results in no loss of information. One
10
O. BERNARDI AND G. CHAPUY
thereby obtains the classical bijection between plane trees and parenthesis systems
on {a, ā}.
Lemma 3.4 (Folklore). The mapping which associates its code to a unicellular
map is a bijection between unicellular map with n edges and unicellular code on the
alphabet An .
Proof. The mapping is injective since the root and the edge-permutation α and
vertex-permutation σ = φα can be recovered from the code. It is also surjective
since starting from any code one obtains a pair of permutation α, σ which indeed
gives a unicellular map T = (An , α, σ) (the only non-obvious property is the transitivity condition, but this is granted by the fact the face-permutation φ = σα is
cyclic).
A word on Ak ⊎ Bl (where Bl = {b1 , b̄1 , . . . , bl , b̄l }) is a code-shuffle if the subwords w|A and w|B made of the letters in Ak and Bl respectively are unicellular
codes on Ak and Bl . Let (M, S) be a covered map, where M = (H, σ, α) and let
k = |S|/2, l = |S̄|/2. By Lemma 3.1, the motion function θ is cyclic. Hence, there
exists a unique way of relabelling the half-edges on the set Ak ⊎ Bl in such a way
that S = Ak , S̄ = Bl , α(ai ) = āi for all i = 1 . . . k, α(bi ) = b̄i for all i = 1 . . . l,
and θ = (w1 , w2 , . . . , w2n ), where w1 is the root of M and w = w1 w2 · · · w2n is a
code-shuffle. We call w the code of the covered map (M, S).
Topologically, the code of a covered map (M, S) is obtained by turning around
the submap T = M|S in counterclockwise direction starting from the root and
writing ai (resp. bi ) when we discover the ith edge in S (resp. S̄) and writing āi
(resp. b̄i ) when we see this edge for the second time. For instance, the code of the
covered map in Figure 7(a) is w = a1 b1 a2 b2 ā2 b3 ā1 b̄1 a3 b4 a4 a5 b̄3 ā5 b̄2 ā4 b̄4 ā3 . We now
state the main result of this preliminary section.
Proposition 3.5. The mapping φ which associates its code to a covered map is a bijection between covered maps with n edges and code-shuffles of length 2n. Moreover,
if w is the code of the covered map (M, S), then w|A is the code of the unicellular
map M|S (on the alphabet A|S|/2 ) and w|B is the code of the dual unicellular map
M|∗S̄ (on the alphabet B|S̄|/2 ).
Proof. To see that φ is injective, observe first that the code-shuffle allows to recover
the root of the map M = (H, σ, α), the subset S = Ak , the edge-permutation α
and the motion function θ = (w1 , . . . , w2n ). From this, the vertex-permutation σ is
deduced by σ(h) = θα(h) if h ∈ S and σ(h) = θ(h) otherwise. We now prove that
φ is surjective. For this, it is sufficient to prove that starting from any shuffle-code,
the pair (M, S) defined as above is a covered map. First note that the permutations
σ and α clearly act transitively on H since θ is cyclic, hence M is a map. Now, the
fact that (M, S) is a a covered map is a consequence of Lemma 3.1 since θ is the
motion function of (M, S) and is cyclic.
′
′′
We now prove the second statement. Let wA = w1′ , . . . , w2k
and wB = w1′′ , . . . , w2l
.
′
′
′′
′′
By definition of restrictions, θ|S = (w1 , . . . , w2k ) and θ|S̄ = (w1 , . . . , w2l ). Moreover, by Proposition 3.1, these restrictions θ|S and θ|S̄ correspond respectively to
the face-permutations of M|S and M|∗S̄ . Recall also that the root r1 of M|S is σ i (r),
where r is the root of M and i is the least integer such that σ i (r) ∈ S. Equivalently,
A BIJECTION FOR COVERED MAPS
11
r1 = θi (r) where i is the least integer such that θi (r) ∈ S, hence r1 = w1′ . Similarly, the root r2 of M|∗S̄ is φj (r) where j is the least integer such that φj (r) ∈ S̄,
or equivalently r2 = θj (r) where j is the least integers such that θj (r) ∈ S̄, hence
r1 = w1′′ . Thus, the words w|A and w|B are the codes of the unicellular maps M|S
and M|∗S̄ respectively.
We now explore the enumerative consequence of Proposition 3.5. Let Ag (n) be
the number of unicellular maps of genus g with n edges. Let Cg1 ,g2 (n1 , n2 ) (resp.
Cg1 ,g2 (n)) be the number of covered maps (M, S) such that the unicellular maps
M|S and M|∗S̄ have respectively n1 and n2 edges (resp. a total of n edges) and genus
+2n2
ways of shuffling unicellular codes of length
g1 and g2 . Since there are 2n12n
1
2n1 and 2n2 , Proposition 3.5 gives
2n1 + 2n2
(9)
Ag1 (n1 )Ag2 (n2 ),
Cg1 ,g2 (n1 , n2 ) =
2n1
and
(10)
Cg1 ,g2 (n) =
n X
2n
Ag1 (m)Ag2 (n − m).
2m
m=0
An alternative equation (used in Section 5) is obtained by fixing the number of
vertices of M|S and M|∗S̄ instead of their genus. Let Av (n) be the number of
unicellular maps with v vertices and n edges (Av (n) = A(n−v+1)/2 (n) by Euler
relation and this number is 0 if n − v + 1 is odd). Let also C v,f (n) be the number
of covered maps with v vertices, f faces and n edges (hence with genus g = (n −
v − f + 2)/2). Proposition 3.5 gives
n X
2n
v,f
(11)
Av (m)Af (n − m).
C (n) =
2m
m=0
Equation (10) generalizes the results used by Mullin [32] and by Lehman and
Walsh [28] in order to count tree-rooted maps. Indeed, the number of tree-rooted
maps of genus g with n edges is
m X
2n
Tg (n) = C0,g (n) =
(12)
Cat(m)Ag (n − m),
2m
n=0
2m
1
where Cat(m) = m+1
is the mth Catalan number. In [32], Mullin proved
m
Equation (7) by applying the Chu-Vandermonde identity to (12) (in the case g = 0).
Similarly, in [28], Lehman and Walsh proved Equation (8) by applying the ChuVandermonde identity to (12) (in the case g = 1). In [2], Bender et al. used the
asymptotic formula
3
1
n3g− 2 n
Ag (n) ∼n→∞ g √ 4 1 + O √
12 g! π
n
which they derived from the expressions given in [27], together with (12) in order
to determine the asymptotic number of tree-rooted maps of genus g and obtained:
Tg (n) ∼n→∞
4
n3g−3 16n .
πg!96g
12
O. BERNARDI AND G. CHAPUY
Applying the same techniques as Bender et al. to Equation (10) gives the asymptotic number of covered maps:
4
g1 + g2
(13)
n3g−3 16n .
Cg1 ,g2 (n) ∼n→∞
πg!96g
g1
In particular, the the total number of covered maps of genus g with n edges satisfies:
(14)
Cg (n) =
g
X
h=0
Ch,g−h (n) ∼
4
n3g−3 16n .
πg!48g
Hence the proportion of tree-rooted maps among covered maps of genus g tends to
1/2g when the size n goes to infinity. We have no simple combinatorial interpretation of this fact.
This concludes our preliminary exploration of covered maps. We now leave the
world of shuffles and concentrate on the main subject of this paper, that is, the
bijection Ψ between covered maps and pairs made of a tree and a unicellular bipartite map.
4. The bijection.
This section contains our main result, that is, the description of the bijection
Ψ between covered maps and pairs made of a tree and a bipartite unicellular map
called the mobile.
Let (M, S) be a covered map. The bijection Ψ consists of two steps. At the first
step, the submap S is used to define an orientation (I, O) of M ; see Figure 7. At
the second step of the bijection, which we call unfolding, the vertices of the map
M are split into several vertices (the rule for the splitting is given in terms of the
orientation (I, O); see Figure 6). The map obtained after these splits is a plane tree
A, and the information about the splitting process can be encoded into a bipartite
unicellular map B. The tree A = Ψ1 (M, S) and the mobile B = Ψ2 (M, S) are represented in Figure 9. We now describe the two steps of the bijection Ψ in more details.
Step 1: Orientation ∆. The orientation step is represented in Figure 7. One
starts with a covered map (M, S) and obtains an oriented map (M, (I, O)). Topologically, the orientation (I, O) is obtained by turning around the submap M|S
(in counterclockwise direction starting from the root) and orient each edge of M
according to the following rule:
• each edge in M|S is oriented in the direction it is followed for the first time
during the tour,
• each edge not in M|S is oriented in such a way that the ingoing half-edge
is crossed before the outgoing half-edge during the tour.
Let us now make definitions precise in terms of the combinatorial definition
of maps. Let (M, S) be a covered map, let r be its root, and let θ be its motion function. Recall from Proposition 3.1 that θ is a cyclic permutation on the
set H of half-edges. Therefore, one obtains a total order ≺S , named appearance
order, on the set H by setting r ≺S θ(r) ≺S · · · ≺S θ|H|−1 (r). Topologically,
the appearance order is the order in which half-edges of M appear when turning
around the spanning submap T = M|S in counterclockwise order starting from the
A BIJECTION FOR COVERED MAPS
13
root. For instance, the order obtained for the spanning submap T in Figure 7(a) is
a1 ≺S b1 ≺S a2 ≺S b2 ≺S ā2 ≺S b3 ≺S · · · ≺S ā3 . We now define the oriented map
(M, (I, O)) = ∆(M, S) which is represented in Figure 7(b).
Definition 4.1. Let (M, S) be a covered map with half-edge set H. The mapping
∆ associates to (M, S) the oriented map (M, (I, O)), where the set I of ingoing
half-edges contains the half-edges h ∈ S such that α(h) ≺S h and the half-edges
h∈
/ S such that h ≺S α(h) (and O = H \ I).
We now characterize the image of the mapping ∆ by defining left-connected
orientations. Let M = (H, σ, α) be a map and let (I, O) be an orientation. Let h0
denote the root of M . A left-path is a sequence h1 , h2 , . . . , hk of ingoing half-edges
such that for all i = 1 . . . k, there exists an integer qi > 0 such that hi−1 = σ qi (α(hi ))
and σ p (α(hi )) ∈ O for all p = 0 . . . qi − 1. In words, a left-path is a directed path
starting from the arrow pointing the root-corner and such that no ingoing halfedges is incident to the left of the path. Clearly, for any ingoing half-edge h, there
exists at most one left-path h1 , h2 , . . . , hk whose extremity is hk is h. We say that an
oriented map (M, (I, O)) is left-connected if every ingoing half-edge is the extremity
of a left-path.
arrow pointing the root-corner
h1
hk
root h0
Figure 5. A left-path.
Theorem 4.2. The mapping ∆ is a bijection between covered maps and leftconnected maps.
The proof of Theorem 4.2 is postponed to Section 6.
Remark on the planar case: It is shown in [4, Prop. 3] that the mapping
∆ is a bijection between planar covered maps (i.e. tree-rooted maps) and planar
oriented maps which are accessible (any vertex can be reach from the root-vertex
by a directed path) and minimal (no directed cycle has the root-face on its right).
Thus, in the planar case the left-connected orientations are the accessible minimal
orientations.
Step 2: Unfolding Λ. The unfolding step is represented in Figures 8 and 9.
One starts with a left-connected map (M, (I, O)) and obtains two maps A =
Λ1 (M, (I, O)) and B = Λ2 (M, (I, O)). The map A is a plane tree and the map
B is a bipartite unicellular map (with black and white vertices). By analogy with
the paper [8], we call the bipartite unicellular B the mobile associated with the
left-connected map (M, (I, O)). Let us start with the topological description of
this step. Let v be a vertex of the oriented map (M, (I, O)) and let h1 , . . . , hd be
the incident half-edges in counterclockwise order around v (here it is important to
consider the arrow pointing the root-corner as an ingoing half-edge). If the vertex
v is incident to k > 0 ingoing half-edges, say hi1 , hi2 , . . . , hik = hd , then the vertex
14
O. BERNARDI AND G. CHAPUY
v of M will be split into k vertices v1 , v2 , . . . , vk of the tree A. The splitting rule is
represented in Figure 6: for j = 1 . . . k, the vertex vj of the trees A is incident to
the half-edges hij−1 +1 , hij−1 +2 , . . . , hij .
h7 h6 h5
h8
h4
h2
h4
h6 h5
v3
v2
v1
h1
h2
h8
v
h1
h7
h3
h3
Figure 6. Splitting of a vertex v incident to 3 ingoing half-edges h3 , h4 , h8 .
Observe that the splitting of the vertex v can be written conveniently in terms
of permutations. Indeed, seeing the vertex v as the cycle (h1 , . . . , hd ) of the vertexpermutation σ and the vertices v1 = (h1 , . . . hi1 ), . . . , vk = (hik−1 +1 , . . . , hik ) as
cycles of the vertex-permutation τ of the tree A gives the following relation between
v and the product of cycles v ′ = v1 v2 . . . , vk (these are both permutations on
{h1 , . . . , hk })
v = v ′ π◦ ,
where π◦ is the permutation such that π◦ (h) = h if h ∈ O and π◦ (hij ) = hij+1
for j = 1, . . . , l. Hence, v ′ = vπ◦−1 , where π◦ = v|I (with the convention that the
restriction v|I acts as the identity on O). The cycle (hi1 , hi2 , . . . , hil ) of π◦ will
represent one of the (white) vertices of the bipartite unicellular map B. This white
vertex is represented in Figure 8(a).
We now describe the unfolding step in more details. Let r be the root of the
map M = (H, σ, α) and let φ = σα be its face-permutation . We consider two
new half-edges i and o not in H and define H ′ = H ∪ {i, o}, I ′ = I ∪ {i} and
O′ = O ∪ {o} (the half-edge i should be thought as this half-edge pointing to the
root-corner, while o should be thought as its dual). We define the involution α′ on
H ′ by setting α′ (i) = o and α′ (h) = α(h) for all h ∈ H. We also define σ ′ as the
permutation on H ′ obtained from σ by inserting the new half-edge i just before the
root r in the cycle of σ containing r and creating a cycle made of o alone (that is,
σ ′ (o) = o). Similarly we define φ′ as the permutation on H ′ obtained from φ by
inserting the new half-edge o just before r in the cycle of φ containing r and creating
a cycle made of i alone. Recall that φ = σα and observe that φ′ = (i, o)σ ′ α′ . We
consider the restrictions
(15)
′
and π• = φ′|O′ .
π◦ = σ|I
′
In the example of Figure 7, one gets π◦ = (i)(ā1 , b1 , b3 )(ā2 , b2 )(ā3 , b4 )(ā5 )(ā4 ) and
π• = (o, a1 , a3 )(b̄1 , a2 , b̄4 , a4 , a5 , b̄3 )(b̄2 ). We now define the permutations π and τ ′
on H ′ , and a permutation τ on H by setting
(16)
′
π = π◦ π•−1 , τ ′ = σ ′ π◦−1 and τ = τ|H
,
′
where a slight abuse of notation is done by considering that π◦ = σ|I
′ acts as the
′
′
′
identity on O and that π• = φ|O′ acts as the identity on I . It is easily seen that
τ ′ (o) = o. On the other hand, we will show (Lemma 6.10) that the half-edge i is not
A BIJECTION FOR COVERED MAPS
ā3 b̄4
b4
b̄4
ā3
ā4
a4
15
b4
b̄2
a5
ā4
a4
b̄2
a5
a3
a3
ā5
b̄1
b̄3
ā1
a1
b3
ā5
b̄1
b2
ā1
a1
ā2
b1 a 2
b̄3
b3
b2
ā2
b1 a 2
Figure 7. (a) A covered map of genus 1 (the unicellular submap
is indicated by thick lines) (b) The associated oriented map.
v2
v3
v
π◦
v1
(a)
π•
(b)
(c)
Figure 8. Representation of the unfolding: (a) around one vertex
(this defines one cycle of π◦ ); (b) around one face (this defines one
cycle of π• ); (c) on the map of Figure 7.
b3
a2
ā2
ā3
b4
b̄3
b̄4 a ā4
4
a2
a5
ā5
b2
b̄2
b4
b̄4
a5
ā4
a4
a3
(a)
i
b̄1
a1
b1
ā3 a b̄1
3
a1
o
ā1
b3
b̄2
b̄3
b2
ā5
ā2
b1
ā1
(b)
Figure 9. The image by Ψ of the covered map (M, S) of Figure 7.
(a) The tree Ψ1 (M, S). (b) The unicellular map Ψ2 (M, S).
alone in its cycle of τ ′ . Hence, the half-edge t = τ ′ (i) is distinct from i and o. We
now consider the pseudo maps A = (H, τ, α) with root t = τ ′ (i) and B = (H ′ , π, α′ )
with root i.
16
O. BERNARDI AND G. CHAPUY
Definition 4.3. We denote by Λ the mapping which to a left-connected map
(M, (I, O)) associates the pair (A, B). We also denote Ψ = Λ ◦ ∆. Lastly if (M, S)
denotes the covered map such that (M, (I, O)) = ∆(M, S), we denote Ψ1 (M, S) =
Λ1 (M, (I, O)) = A and Ψ2 (M, S) = Λ2 (M, (I, O)) = B.
The images (A, B) of the covered map in Figure 7(a) by the mappings Ψ1 and
Ψ2 are represented respectively in Figure 9(a) and (b). In terms of permutations,
one gets A = (H, τ, α) and B = (H ′ , π, α′ ), where
τ = (a1 , b̄1 , ā3 )(ā1 )(b1 )(ā3 , a4 , b̄4 )(ā4 , a5 , b2 )(ā5 , b̄3 )(b3 , a2 )(ā2 )(b̄2 )(b4 )
and
π = (i)(ā1 , b1 , b3 )(ā2 , b2 )(ā3 , b4 )(ā4 )(o, a3 , a1 )(b̄3 , a5 , a4 , b̄4 , a2 , b̄1 )(b̄2 ).
Our main result is the following theorem which will be proved in Section 6.
Theorem 4.4. The mapping Ψ = Λ ◦ ∆ which to a covered map (M, S) associates
the pair (Ψ1 (M, S), Ψ2 (M, S)) is a bijection between covered maps of size n and
genus g and pairs made of a plane tree Ψ1 (M, S) of size n and a bipartite unicellular
map Ψ2 (M, S) of size n+1 and genus g. Moreover by coloring the vertices of the
bipartite map Ψ2 (M, S) in two colors, say white and black, with the root-vertex
being white, one gets v(M ) white vertices and f (M ) black vertices.
Remark (topological intuition). From Figures 8(a) and (b), the reader should
see that the mobile B = Λ2 (M, (I, O)) has white vertices (the cycles of π◦ made of
half-edges in I ′ ) corresponding to the vertices of M and black vertices (the cycles
of π•−1 made of half-edges in O′ ) corresponding to the faces of M . The topological
intuition explaining that A = Ψ1 (M, S) is connected, is that left-paths are preserved during the unfolding step. It then follows that A is a plane tree because its
number of vertices is one more than its number of edges. The topological intuition
explaining that the mobile B is a unicellular map comes from the fact that A can
reach every white corners of B (without crossing its edges). Indeed, this implies
that the pseudo-map B has no contractible cycles. Using this fact and the relation
between the number of vertices and edges shows that B is a unicellular map of
genus g(M ).
5. Enumerative corollaries
In this section we explore the enumerative consequences of the bijection Ψ, and
establish the equivalence between the formula (5) of Harer and Zagier and the
formula (6) of Jackson.
Recall the notations of Section 3: Ag (n), Bg (n), Cg (n) are respectively the
number of general unicellular maps, bipartite unicellular maps, and covered maps
with n edges and genus g. Similarly Av (n), B v,f (n), C v,f (n) are respectively the
number of general unicellular maps with v vertices, bipartite unicellular maps with
v white and f black vertices, and covered maps with v vertices and f faces having
n edges. The first direct consequence of Theorem 4.4 is:
Theorem 5.1. The numbers Cg (n) and C v,f (n) of covered maps, and the numbers
Bg (n) and B v,f (n) of bipartite maps are related by:
(17)
(18)
C
Cg (n)
v,f
=
(n) =
Cat(n) Bg (n + 1)
Cat(n) B v,f (n + 1)
A BIJECTION FOR COVERED MAPS
where Cat(n) =
17
2n
1
n+1 n
.
Observe that for g = 0 we have B0 (n+1) = Cat(n+1), so that Equation (17) coincides with Mullin’s formula (Equation (7)). Using known closed-form expressions
for the numbers Bg (n) (see [19]), one obtains similar formulas for other genera. For
example in genus 1 and 2 we obtain the following expressions for the number of
covered maps with n edges:
C1 (n) = Cat(n)
(2n − 1)!
(5n2 + 3n + 4)(2n)!
, C2 (n) = Cat(n)
.
6(n − 2)!(n − 1)!
1440(n − 1)!(n − 4)!
We now examine the special case of the torus (genus 1). By Lemma 3.3, a covered
map on the torus is either a tree-rooted map (the submap has genus 0, that is, is a
spanning tree) or the dual of a tree-rooted map. Since duality is involutive, exactly
half of toroidal covered maps of given size are tree-rooted maps. This gives the
first bijective proof to the following result:
Corollary 5.2 (Lehman and Walsh [28]). The number of tree-rooted maps with n
edges on the torus is:
T1 (n) =
1
(2n)!(2n − 1)!
C1 (n) =
.
2
12(n + 1)!n!(n − 1)!(n − 2)!
Another enumerative byproduct of our bijection is a relation between the numbers of general and bipartite unicellular maps. Indeed, by comparing the expression
of C v,f (n) obtained by the shuffle approach (Equation (11)) with the one of Theorem 5.1, we obtain the following:
Theorem 5.3. The numbers of bipartite and monochromatic unicellular maps are
related by the formula:
X
n!(n + 1)! v
(19)
B v,f (n + 1) =
A (n1 )Af (n2 )
(2n
)!(2n
)!
1
2
n +n =n
1
2
Moreover, this formula establishes the equivalence between the formula (5) due to
Harer and Zagier and the formula (6) due to Jackson.
Proof. The first statement is obtained by comparing Equations (11) and (18). We
now show that (5) implies (6). Using (5) one gets:
X
B p,q (n + 1)y p z q
p,q≥1
Eq. (19)
=
=
Eq. (5)
=
=
n!(n + 1)! p
A (n1 )Aq (n2 )y p z q
(2n
)!(2n
)!
1
2
p,q≥1 n1 +n2 =n



X
X
X
n!(n + 1)! 
Ap (n1 )y p  
Aq (n2 )z q 
(2n
)!(2n
)!
1
2
n1 +n2 =n
p≥1
q≥1
X n!(n + 1)! X
y
z
n2
n1
i+j−2
2
n
2 n1 !n2 !
i
i−1 j−1
j
n1 +n2 =n
i,j≥1
X 2i+j−n−2 n!(n + 1)! y
X
1
z
(i − 1)!(j − 1)!
i
j
(n
−
i
+
1)!(n
1
2 − j + 1)!
n +n =n
X
i,j≥1
X
1
2
n1 ≥i−1, n2 ≥j−1
18
O. BERNARDI AND G. CHAPUY
where the second and fourth equalities just correspond to rearrangements of the
summations. Moreover, the inner sum in the last equation is equal to 2n−i−j−2 /(n−
i − j + 2)! by Newton’s binomial theorem. This gives Jackson’s formula (6).
Conversely, observe that the numbers Ap (n) are uniquely determined by the value of
P
n!(n+1)!
Ap (n1 )Ap (n2 ) for all p, n (by induction on n). Thus,
the sums n1 +n2 =n (2n
1 )!(2n2 )!
the calculations above show that Jackson’s formula (6) implies the Harer-Zagier
formula (5).
Remark. Connoisseurs know that formulas (5) and (6) can be interpreted in terms
of unicellular maps with colored vertices (see e.g. [22, sec. 3.2.7]). For those readers, we point out that the equivalence between Harer-Zagier and Jackson’s formulas
can be seen in terms of colorings as well: Ψ is a bijection between shuffles of two
unicellular maps with vertices colored using all colors respectively in {1, . . . , i} and
{1, . . . , j} and pairs made of a plane tree and a unicellular map with black and
white vertices colored using all colors respectively in {1, . . . , i} and {1, . . . , j}.
6. Proofs and inverse bijection.
This section is devoted to the proof of Theorems 4.2 and 4.4 concerning respectively the orientation and unfolding steps of the bijection Ψ.
6.1. Proofs concerning the orientation step. In this Subsection, we prove
Theorem 4.2 about the orientation step ∆ and and define the inverse mapping Γ.
Given an oriented map (M, (I, O)) with vertex-permutation σ and face-permutation
φ, the backward function β is defined by β(h) = σ(h) if h ∈ O and β(h) = φ(h)
otherwise. We point out that the backward function is not a permutation, since
β(h) = β(α(h)) for any half-edge h.
Lemma 6.1. Let (M, (I, O)) be an oriented map with root h0 and let β be the
backward function. The oriented map (M, (I, O)) is left-connected if and only if for
any half-edge, there exists an integer q > 0 such that β q (h) = h0 .
Proof. Suppose first that (M, (I, O)) is left-connected. Let h be a half-edge. If
h is ingoing, then it is the extremity of a left path h1 , . . . , hk = h. By definition
of left-paths, there exist positive integers q1 , . . . , qk such that hi−1 = β qi (hi ) for
all i = 1 . . . k. Hence, β q (h) = h0 for q = q1 + · · · + qk . Now, if h is outgoing,
β(h) = β(α(h)), hence there exists q > 0 such that h0 = β q (α(h)) = β q (h).
Suppose conversely that for any half-edge h, there exists an integer q > 0
such that β q (h) = h0 . In this case, for any ingoing half-edge h, the sequence
h1 , h2 , . . . , hk = h of ingoing half-edges appearing (in this order) in the sequence
β q−1 (h), β q−2 (h), . . . , β(h), h is a left-path. Hence (M, (I, O)) is left-connected. Proposition 6.2. The image of any covered map by the mapping ∆ is left-connected.
Proof. Let (M, S) be a covered map, where the map M = (H, σ, α) has root r, and
let (M, (I, O)) be its image by ∆. Our strategy is to prove that for any half-edge h
such that β(h) 6= r, one has h ≺S β(h). This will clearly prove that the sequence
β(h), β 2 (h), β 3 (h) . . . must contain the root r, and by Lemma 6.1, that (M, (I, O))
left-connected.
We distinguish four cases, depending on the fact that h is in I or O, and in S or
S̄. In these four cases, we denote h′ the half-edge θ−1 (β(h)), where θ is the motion
A BIJECTION FOR COVERED MAPS
19
function of (M, S). Observe that h′ ≺S β(h) since by hypothesis β(h) 6= r.
Case 1: h is in O and in S̄. In this case, one has β(h) = σ(h) = θ(h), hence h′ = h.
Thus h = h′ ≺S β(h).
Case 2: h is in O and in S. In this case, one has β(h) = σ(h) = θ(α(h)), hence
h′ = α(h). Moreover, by definition of ∆, one has h ≺S α(h) thus h ≺S h′ ≺S β(h).
Case 3: h is in I and in S̄. In this case, one has β(h) = σ(α(h)) = θ(α(h)), hence
h′ = α(h). Moreover, by definition of ∆, one has h ≺S α(h), thus h ≺S h′ ≺S β(h).
Case 4: h is in I and in S. In this case, one has β(h) = σα(h) = θ(h), hence
h′ = h. Thus, h = h′ ≺S β(h).
We will now define a mapping Γ that we will prove to be the inverse of ∆. Let
us first give the intuition behind the injectivity of ∆ by considering a covered map
(M, S) with motion function θ and its image (M, (I, O)) by ∆. Observe from the
definition of ∆ that the root r of M is in S if and only if it is in O. Thus, it is
possible to know from the orientation (I, O) whether r belongs to S or not, and
thereby deduce the next half-edge h1 = θ(r) around the submap M|S . The same
reasoning will allow to determine, from the orientation (I, O), whether the halfedge h1 belongs to S and deduce the next half-edge h2 = θ(h1 ) around M|S and so
on. . . This should convince the reader that the mapping ∆ is injective and highlight
the definition of Γ given below.
It is convenient to define Γ as a procedure which given an oriented map (M, (I, O))
with root r returns a subset S of half-edges. This procedure visit some half-edges of
M starting from the root r, and decide at each step whether the current half-edge
h belongs to the set S or not.
Definition 6.3. The mapping Γ associates to an oriented map (M, (I, O)) the
spanned map obtained by the following procedure.
Initialization: Set S = ∅, R = ∅ and set the current half-edge h to be the root r.
Core:
• If h ∈
/ S ∪ R do:
If h is in O then add h and α(h) to S; otherwise add h and α(h) to R.
• Set the the current half-edge h to be σα(h) if h in S and σ(h) otherwise.
Repeat until the current half-edge h returns to be r.
End: Return the spanned map (M, S).
We first prove the termination of the procedure Γ.
Lemma 6.4. For any oriented map (M, (I, O)), the procedure Γ terminates and
returns a spanned map (M, S). Moreover, the list of all successive current half-edges
visited by the procedure is the cycle containing the root r of the motion function θ
associated to (M, S).
Proof. We first prove that the procedure Γ terminates. Observe that, at any step
of the procedure, the sets S and R are disjoint and stable by α. Moreover, these
sets are both increasing, hence they are constant after a while, equal to some sets
S∞ and R∞ which are disjoint. Let θ∞ be the motion function of the submap
M|S∞ . Then, at each core step of the procedure, the current half-edge h becomes
the half-edge θ∞ (h). Indeed, if at the current step h is in S, then h is in S∞ (since
the set S cannot decrease) so that θ∞ (h) = σα(h), while if h is in R, then h is in
R∞ (since the set R cannot decrease), hence it is not in S∞ so that θ∞ (h) = σ(h).
20
O. BERNARDI AND G. CHAPUY
Hence the sequence of all successive current half-edges form a cycle of the permutation θ∞ . Since the procedure starts with h equal to the root r, it follows that
r is reached a second time, and that the procedure terminates. Finally, the spanned
map returned by the algorithm is (M, S∞ ), which concludes the proof.
Proposition 6.5. The image of a left-connected map by the mapping Γ is a covered
map.
Proof. Let (M, (I, O)) be a left-connected map. We denote by r the root of M =
(H, σ, α), by β the backward function of (M, (I, O)), and by θ the motion function
of (M, S) = Γ((M, (I, O))). Let K be the set of half-edges contained in the cycle
of the motion function θ containing the root r. In order to prove the proposition,
it suffices to prove that K = H. Indeed, in this case the motion function θ is cyclic
which implies that (M, S) is a covered map by Proposition 3.1.
Moreover, since (M, (I, O)) is left-connected, for any half-edge h in H, there
exists a positive integer q such that β q (h) is the root r which belongs to K. Hence,
it suffices to prove that any half-edge h such that β(h) is in K, is also in K.
Let h be an half-edge such that β(h) is in K and let h′ = θ−1 (β(h)). Observe
that, by definition of K, the half-edge h′ is in K. We now distinguish four cases,
depending on the fact that h is in I or O, and in S or S̄.
Case 1: h is in O and in S̄. In this case, one has β(h) = σ(h) = θ(h), hence h′ = h.
Thus h = h′ is in K.
Case 2: h is in O and in S. In this case, by definition of Γ, the half-edge h was
the current half-edge when it was added to S. Thus, by Lemma 6.4, the half-edge
h is in K.
Case 3: h is in I and in S̄. In this case, one has β(h) = σ(α(h)) = θ(α(h)), hence
h′ = α(h). Since h′ is in K, Lemma 6.4 ensures that it was the current half-edge
at a certain step of the procedure Γ. Hence, since h′ = α(h) is in O but not in S,
it means that h and h′ were added to the set R at a step of the procedure Γ, such
that h was the current half-edge. Thus, by Lemma 6.4, the half-edge h is in K.
Case 4: h is in I and in S. In this case, one has β(h) = σα(h) = θ(h), hence
h′ = h. Thus h = h′ is in K.
We now complete the proof of Theorem 4.2.
Proposition 6.6. The mappings ∆ and Γ are inverse bijections between covered
maps and left-connected maps.
Proof. We first prove that the mapping ∆◦Γ is the identity on left-connected maps.
Observe that this composition is well defined by Proposition 6.5. Let (M, (I, O))
be a left-connected map and let (M, S) be its image by Γ. We want to prove that
˜ Õ)) ≡ ∆(M, S) is equal to (M, (I, O)). For that, it suffices to show that
(M, (I,
any half-edge h such that h ≺S α(h) is in O if and only if it is in Õ. Let h be such
a half-edge. By definition of ∆, it follows that h is in Õ if and only if h is in S.
Now, by Lemma 6.4, the sequence h1 = r, h2 , . . . , h2n of current half-edge visited
during the procedure Γ satisfies h1 ≺S h2 ≺S · · · ≺S h2n , hence h is visited before
α(h) during the procedure Γ. Hence, by definition of Γ, the half-edge h is in O if
and only if h is in S.
We now prove that the mapping Γ ◦ ∆ is the identity on covered maps. Observe
that this composition is well defined and returns a covered map by Propositions 6.2
and 6.5. Let (M, S) be a covered map with root r and let (M, (I, O)) be its image
A BIJECTION FOR COVERED MAPS
21
by ∆. Let also (M, S ′ ) = Γ(M, (I, O)) and let θ and θ′ be respectively the motion
function of (M, S) and (M, S ′ ). In order to prove that S = S ′ it suffices to prove that
θ = θ′ (indeed, the set S and S ′ are completely determined by θ and θ′ ). Suppose
k+1
now that θ 6= θ′ and consider the smallest integer k ≥ 0 such that θk+1 (r) 6= θ′
(r)
′
(such an integer exists since θ and θ are cyclic). Observe that for all 0 ≤ j < k, the
j
j+1
half-edge θj (r) = θ′ (r) is in S if and only if it is in S ′ (since θj+1 (r) = θ′ (r)).
k
On the other hand, the half-edge h = θk (r) = θ′ (r) is in the symmetric difference of
′
k+1
′ k+1
S and S (since θ
(r) 6= θ
(r)). This implies that h ≺S α(h) and h ≺S ′ α(h).
Since h ≺S α(h), the definition of ∆ shows that the half-edge h is in S if and only if
h is in O. Since h ≺S ′ α(h), Lemma 6.4 proves that h is the current half-edge before
α(h) in the procedure Γ on (M, (I, O)). Hence, by definition of Γ, the half-edge h
is in S ′ if and only if h is in O. This proves that h = θk (r) is not in the symmetric
difference of S and S ′ , a contradiction.
Before leaving the world of left-connected maps, we prove the following result.
Lemma 6.7. If (M, (I, O)) is a left-connected map with root r, then every non-root
vertex of M is incident to a half-edge in I and every non-root face is incident to a
half-edge in O.
Proof. If a cycle of the vertex-permutation σ contains no edge in I, then β(h) =
σ(h) for every half-edge h in this cycle. By Lemma 6.1, this implies that the root
r belongs to this cycle. Similarly, if a cycle of the face-permutation φ contains no
edge in O, then β(h) = φ(h) for every half-edge h in this cycle. By Lemma 6.1,
this implies that the root r belongs to this cycle.
6.2. Proofs concerning the unfolding step. In this Subsection we prove Theorem 4.4. We fix a left-connected map (M, (I, O)), where the map M = (H, σ, α)
has n edges, genus g, root r and face-permutation φ. We denote A = (H, τ, α) =
Λ1 (M, (I, O)) and B = (H ′ , π, α′ ) = Λ2 (M, (I, O)) and adopt the notation of Section 4 for the sets H ′ , I ′ , O′ and the permutations σ ′ φ′ , τ ′ , π◦ and π• .
Lemma 6.8. The permutations τ, α act transitively on H, thus A is a map.
As mentioned above, the intuition behind the connectivity of A is that left-paths
are preserved by the unfolding. This can be formalized as follows.
Proof. Let β be the backward function of the oriented map (M, (I, O)) defined by
β(h) = σ(h) if h is in O and β(h) = σα(h) otherwise. Let β̃ be the backward
function of the oriented map (A, (I, O)) defined by β̃(h) = τ (h) if h is in O and
β̃(h) = τ α(h) otherwise.
We first prove that β(h) = β̃(h) for any half-edge h ∈ H such that β(h) 6= r. If
h is in O (and β(h) 6= r), then
β(h) ≡ σ(h) = σ ′ (h) = τ ′ (h) = τ (h) ≡ β̃(h),
since the permutations σ ′ and τ ′ coincide on O′ . If h is in I (and β(h) 6= r), then
β(h) ≡ σ(α(h)) = σ ′ (α(h)) = τ ′ (α(h)) = τ (α(h)) ≡ β̃(h),
since again the permutations σ ′ and τ ′ coincide on O′ .
Since the oriented map (M, (I, O)) is left-connected, Lemma 6.1 ensures that for
any half-edge h, there exists an integer q > 0 such that β q (h) = r. Taking the least
22
O. BERNARDI AND G. CHAPUY
such integer q, and using the preceding point shows that for any half-edge h, there
exists an integer q > 0 such that β̃ q−1 (h) = β q−1 (h) = l, where l is a half-edge
such that β(l) = r. Moreover, the relation β(l) = r shows that l is either equal to
u = σ −1 (h) or to α(u). Therefore, any half-edge h can be sent to one of the halfedges u, α(u) by applying the function β̃, hence by acting with the permutations τ
and α. This proves the lemma.
We call root-to-leaves orientation of a plane tree the unique orientation such that
each non-root vertex is incident to exactly one ingoing half-edge.
Proposition 6.9. The map A is a tree. Moreover, (I, O) is the root-to-leaves
orientation of A.
We start with an easy lemma.
Lemma 6.10. The half-edge u = σ −1 (r) is in O and τ ′ (u) = i. In particular, the
half-edge i is not alone in its cycle of τ ′ .
Proof. Consider the backward function β of the oriented map (M, (I, O)). Since,
(M, (I, O)) is left-connected, Lemma 6.1 ensures that there exists a half-edge h such
that β(h) = r. Since β(h) = β(α(h)) we can take h in O and get h = σ −1 (r) = u.
This proves that u = σ −1 (r) is in O. It is then obvious from the definition of τ ′
that τ ′ (u) = i.
Proof of Proposition 6.9. Recall that n = |H|/2 denotes the number of edges of
M , hence of A. We first prove that the map A has (at least) n + 1 vertices. By
−1
construction, the permutation τ ′ = σ ′ π◦−1 = σ ′ σ ′ |I has at most one element of I ′
in each of its cycle. Hence it has at least n + 1 cycles beside the cycle made of o
alone. Moreover, by Lemma 6.10, the half-edge i is not alone in its cycle of τ ′ , thus
′
has at least n + 1 cycles.
the vertex-permutation τ = τ|H
The (connected) map A has n edges and (at least) n + 1 vertices hence it is a
tree. Moreover, each non-root vertex of A is incident to exactly one half-edge in I.
Thus, (I, O) is the root-to-leaves orientation of A.
We prove a last easy lemma about the tree A = (H, τ, α).
Lemma 6.11. The permutation τ ′ = σ ′ π◦−1 is obtained from the vertex-permutation
τ by inserting the half-edge i before the root t of A in the cycle of τ containing t
and creating a cycle made of o alone. The permutation ϕ′ = (i, o)τ ′ α′ is obtained
from the face-permutation ϕ = τ α by inserting the half-edge o before t in the cycle
of ϕ containing t and creating a cycle made of i alone.
′
= τ . Moreover, it is easy to check that
Proof. By definition, τ ′ (i) = t and τ|H
′
′ −1
τ (o) = σ π◦ (o) = o. This proves the statement about τ ′ . We now denote u =
−1
τ ′ (i). By definition, ϕ(α(u)) = τ (u) = t and one can check ϕ′ (α(u)) = o,
ϕ′ (o) = t, ϕ′ (i) = i, and ϕ′ (h) = ϕ(h) for all h ∈
/ {i, o, α(u)}. This proves the
statement about ϕ′ .
We will now prove that the mobile B = (H ′ , π, α′ ) = Λ2 (M, (I, O)) is a unicellular bipartite map. We introduce some notations for the half-edges in H. From
Proposition 6.9, the tree (A, (I, O)) is oriented from root to leaves. In particular,
the root t of A is outgoing. We denote o1 = t, o2 , . . . , on the outgoing half-edges as
appearing during a counterclockwise tour around the tree A, that is to say, ϕ|O =
A BIJECTION FOR COVERED MAPS
23
(o1 , o2 , . . . , on ) where ϕ = τ α is the face-permutation of A. This labelling is indicated in Figure 10(a). Observe that Lemma 6.11 implies ϕ′ |O′ = (o, o1 , o2 , . . . , on ).
For j = 1, . . . , n, we denote ij = α(oj ), so that α′ ϕ′ |O α′ = (i, i1 , i2 , . . . , in ).
o3
i3
o4
o5
A
o3
i2
i4
i6
o6
i5
i3 i
4
i1
o2
i1
o1
o4
i6
o6
o1 o i2
2
o5
i5
B
o
i
Figure 10. A pair (A, B) coherently labelled on the alphabet
{i, i1 , . . . , i6 , o, o1 , . . . , o6 }: the face-permutation ϕ of the tree A
satisfies ϕ′|O′ = (o, o1 , . . . , o6 ), while the face-permutation ψ of the
mobile B satisfies ψ|I ′ = (i, i1 , . . . , i6 ).
Proposition 6.12. The mobile B = Λ2 (M, (I, O)) is a bipartite unicellular map
of genus g(M ). Moreover, if we color the vertices of B in two colors, say white and
black, with the root-vertex being white, then it has v(M ) white vertices and f (M )
black vertices. Lastly, the half-edges incident to white vertices are i, i1 , i2 , . . . , in
−1
and appear in this order during a clockwise tour around B, that is to say, ψ|I
′ =
′ ′
′
′
(i, i1 , . . . , in ) = α ϕ |O α , where ψ = πα is the face-permutation of B and ϕ′ =
(i, o)τ ′ α′ .
−1
Lemma 6.13. The permutation ψ = πα′ and ϕ′ = (i, o)τ ′ α′ are related by ψ|I
′ =
(i, i1 , . . . , in ) = α′ ϕ′ |O′ α′ .
−1
Proof. We consider a half-edge h in I ′ and want to show α′ ϕ′ |O′ α′ (h) = ψ|I
′ (h).
′
′
′
′
First observe that for any permutation γ on H , one has α γ|O′ α = (α γα′ )I ′
because the involution α′ maps I ′ to O′ . In particular, α′ ϕ′ |O′ α′ = (α′ ϕ′ α′ )|I ′ ≡
−1
(α′ (i, o)σ ′ σ ′ |I ′ )|I ′ . Thus,
−1
−1
α′ ϕ′ |O′ α′ (h) = (α′ (i, o)σ ′ σ ′ |I ′ )|I ′ (h) = (α′ (i, o)σ ′ )|I ′ σ ′ |I ′ (h),
−1
since σ ′ |I ′ acts as the identity on O′ .
−1
−1
We now determine ψ|I
≡ α′ π −1 maps I ′ to
′ (h). Observe that permutation ψ
′
′
′
−1
O (since I and O are stable by π ≡ π◦ π• ). Thus,
−1
−1 −1
ψ|I
ψ (h) ≡ α′ π −1 α′ π −1 (h) = α′ π• α′ π◦−1 (h).
′ (h) = ψ
′
By definition, π• = φ′|O′ , φ′ = (i, o)σ ′ α′ and π◦ = σ|I
′ , hence
−1
−1
−1
′ ′ ′
′
′
α′ π• α′ π◦−1 (h) ≡ α′ φ′|O′ α′ σ|I
′ (h) = (α φ α )|I ′ σ|I ′ (h) ≡ (α (i, o)σ )|I ′ σ|I ′ (h).
This gives
−1
−1
′
′
′ ′
′
ψ|I
′ (h) = (α (i, o)σ )|I ′ σ|I ′ (h) = α ϕ |O ′ α (h)
which concludes the proof.
24
O. BERNARDI AND G. CHAPUY
Proof of Proposition 6.12. By Lemma 6.13, every half-edge in I ′ belongs to the
same cycle C of the permutation ψ = πα′ . Now, if h is a half-edge in O′ , ψ(h) =
πα(h) is in I ′ (since I ′ is stable by π) hence it belongs to the cycle C. Thus,
the permutation ψ is cyclic. Hence, the permutations π and α act transitively on
H ′ , that is, B is a map. Moreover, its face-permutation ψ is cyclic, that is, B is
unicellular.
Moreover, since the sets I ′ and O′ are stable by the vertex-permutation π and
exchanged by the edge-permutation α′ , the map B is bipartite. Let us therefore
consider the bipartite coloring where the vertices incident to half-edges in I ′ are
white while the vertices incident to the half-edges in O′ are black. By Lemma 6.7,
each of the cycles of σ ′ except the cycle made of o alone contains at least one half′
′
edge in I ′ . Therefore, the number of cycles of the permutation π◦ = σ|I
′ on I is
the number v(M ) of cycles of the vertex-permutation σ. Similarly, the number of
cycles of the permutation π• = φ′|O′ on O′ is the number f (M ) of cycles of the
face-permutation φ. Thus, the map B has v(M ) white vertices and f (M ) black
vertices. Now, Euler relation gives
2g(B) = 2 + e(B) − f (B) − v(B) = 2 + (e(M ) + 1) − 1 − (v(M ) + f (M )) = 2g(M ).
Thus, the genus of B is g(M ). This concludes the proof of Proposition 6.12.
Topological description of the folding step (Figure 11). We now define a
mapping Ω, the folding step, which we will prove to be the inverse of the unfolding
step Λ. Before defining Ω in terms of permutations, let us explain the topological interpretation of Proposition 6.12. We denote by v0 , v1 , . . . , vn the vertices
of A in counterclockwise order around A (starting from the root-corner) and by
x0 , x1 , . . . , xn the first corners of these vertices; see Figure 11(a). Equivalently, v0
is the root-vertex and x0 is the root-corner, while for j = 1 . . . n, vj is the vertex incident to the ingoing half-edge ij and xj is the corner following ij in counterclockwise
order around vj ; see Figure 10. We also denote by y0 , . . . , yn the white corners of
B in clockwise order around B (starting from the root-corner y0 ); see Figure 11(a).
By Proposition 6.12, for j = 0 . . . n, yj is the corner of B following the half-edge ij
in clockwise order around the incident vertex. Therefore this proposition indicates
how the folding step Ω = Λ−1 should be defined topologically: for j = 0, . . . , n the
first-corner xj of the vertex vj (the jth vertex of A in counterclockwise direction)
is glued to the corner yj (the jth white corner of B in clockwise direction). This
gives a map containing edges of both A and B that we call partially folded map
which is represented in Figure 11(b). The oriented map (M, (I, O)) = Ω(A, B) is
obtained from the partially folded map by keeping the half-edges of A with their
cyclic ordering around the white vertices of B (while the edges and black vertices
of B are deleted).
We now defines the mapping Ω in terms of permutations. Let à = (H, τ̃ , α) be a
rooted plane tree with n = |H|/2 edges, and B̃ = (H ′ , π̃, α′ ) be a rooted bipartite
unicellular map with n + 1 = |H ′ |/2 edges. We consider the usual black-and-white
coloring of B̃ (with the root-vertex being white). The pair (Ã, B̃) is said coherently
labelled if the following conditions are satisfied:
(i) H ′ = H ∪ {i, o}, where i is the root of M and o = α′ (i).
(ii) α = α′|H .
A BIJECTION FOR COVERED MAPS
x3
x6
x4
x5
x2
25
y3
y1
y2
y6
y4
x1
y5
x0
y0
(a)
(b)
(c)
Figure 11. Topological representation of the folding step for the
pair (A, B). Figure (a) indicates the correspondence between
the first corners x0 , . . . , xn of the tree A and the white corners
y0 , . . . , yn of the mobile B. Figure (b) represents the partially
folded map. Figure (c) represents the oriented map (M, (I, O)) =
Ω(A, B).
(iii) The root-to-leaves orientation (I, O) of à is such that the half-edges in I ′ =
I ∪ {i} are incident to white vertices of B̃, while half-edges in O′ = O ∪ {o}
are incident to black vertices of B.
(iv) If the half-edges o1 , o2 , . . . , on in O appear in this order in counterclockwise direction around à with o1 being the root of Ã, then the half-edges
i, i1 , . . . , in defined by ij = α(oj ) for j = 1 . . . n appear in this order in clock−1
′ ′
′
wise direction around B̃. Equivalently, ψ|I
′ = (i, i1 , . . . , in ) = α ϕ |O α ,
where ψ = πα′ is the face-permutation of B and ϕ′ = (i, o)τ ′ α′ .
For example, the pair (A, B) represented in Figure 10 is coherently labelled. Observe that Proposition 6.12 precisely states that the image by Λ of a left-connected
map (M, (I, O)) is coherently labelled. We now prove (the somewhat obvious fact)
that any pair (Ã, B̃) can be relabelled coherently.
Lemma 6.14. Let à = (H, τ̃ , α) be a rooted plane tree with n = |H|/2 edges, and
B̃ = (H ′ , π̃, α′ ) be a rooted bipartite unicellular map with n + 1 = |H ′ |/2 edges.
Then, there is a unique way of relabelling B̃ in such a way that the pair (Ã, B̃) is
coherently labelled.
Proof. We denote by (I, O) the root-to-leaves labelling of A. We denote by I ′
and O′ respectively the set of half-edges incident to white and black vertices of B̃
and observe that these sets are exchanged by α′ (since B̃ is bipartite). We denote
ϕ̃ = τ̃ α and ψ̃ = π̃α′ are the face-permutation of à and B̃ respectively. Lastly, we
denote (o1 , o2 , . . . , on ) the cycle ϕ̃|O with o1 being the root of à and (i′ , i′1 , . . . , i′n )
−1
′
the cycle ψ̃|I
′ with i being the root of B̃. Now we consider the relabelling of B̃
given by the bijection λ from H ′ to H ∪ {o, i} (where i, o are half-edges not in
H) given by λ(i′ ) = i, λ(α(i′ )) = o and for all j = 1 . . . n, λ(i′j ) = α(oj ) and
λ(α(i′j )) = oj . Clearly λ is a bijection and it is the unique bijection making the
pair (Ã, B̃) coherently labelled.
We denote by Pn the set of pairs (Ã, B̃) made of a tree of size n and a unicellular
bipartite map of size n+1. We now consider such a pair (Ã, B̃), where à = (H, τ̃ , α)
and B̃ = (H ′ , π̃, α′ ). By Lemma 6.14, we can assume that the pair (Ã, B̃) is
26
O. BERNARDI AND G. CHAPUY
coherently labelled and we adopt the notations i, o, I, O, I ′ , O′ introduced in the
conditions (i- iv) (in particular, (I, O) is the root-to-leaves orientation of A). We
then define τ̃ ′ as the permutation on H ′ obtained from τ by inserting i before the
root t of à in the cycle containing it and creating a cycle made of o alone. We also
define the permutations π̃◦ and σ̃ ′ on H ′ and the permutation σ̃ on H by
(20)
′
,
π̃◦ = π̃|I ′ , σ̃ ′ = τ̃ ′ π̃◦ and σ̃ = σ̃|H
(where a slight abuse of notation is done by considering π̃◦ as a permutation on
H ′ acting as the identity on O′ ). With these notations, we define Ω(Ã, B̃) =
(M̃ , (I, O)), where M̃ = (H, σ̃, α).
We now complete the proof of Theorem 4.4 by proving the following proposition.
Proposition 6.15. The mappings Λ and Ω are inverse bijections between leftconnected maps of size n and pairs in Pn .
Proof. • We first prove that the mapping Ω ◦ Λ is the identity on left-connected
maps.
Let (M, (I, O)) be a left-connected map, where M = (H, σ, α). Let (A, B) =
Λ(M, (I, O)), where A = (H, τ, α) and B = (H ′ , π, α′ ) (recall that (A, B) is co˜ Õ)) = Ω(A, B), where
herently labelled by Proposition 6.12). Let also (M̃ , (I,
˜ Õ) and M = M̃ (or equivaM̃ = (H, σ̃, α). We want to prove that (I, O) = (I,
lently, σ = σ̃).
˜ Õ) is the root-to-leaves orientation of A. Moreover, by
By definition of Ω, (I,
Proposition 6.9, (I, O) is also the root-to-leaves orientation of A. Hence, (I, O) =
˜ Õ).
(I,
′
′
, where σ ′ = τ ′ π◦ = τ ′ π|I ′ (see (16)). Similarly, σ̃ = σ̃|H
By definition, σ = σ|H
′
′
′
′
and σ̃ = τ̃ π̃◦ = τ̃ π|I ′ (see (20)). Moreover, Lemma 6.11 ensures that τ = τ̃ ′
(since the permutations τ ′ and τ ′′ are obtained from τ by the same procedure).
Thus, σ ′ = σ̃ ′ and σ = σ̃.
• We now prove that the mapping Λ ◦ Ω is the identity on Pn .
We must first prove that this mapping is well defined, that is, the image of any
pair (Ã, B̃) ∈ Pn by Ω is a left-connected map. Let us denote à = (H, τ̃ , α) and
B̃ = (H ′ , π̃, α′ ). By Lemma 6.14, we can assume that the pair (Ã, B̃) is coherently
labelled and we adopt the notations i, o, I, O, I ′ , O′ of conditions (i-iv) and the
notations π̃◦ , σ̃ ′ , σ̃ introduced in (20). Lastly, we denote Ω(Ã, B̃) = (M̃ , (I, O)),
where M̃ = (H, σ̃, α).
Let β̃ be the backward function of the tree à defined on H by β̃(h) = τ̃ (h) if h
is in O and β̃(h) = τ̃ α(h) otherwise. Let β be defined on H by β(h) = σ̃(h) if h is
in O and β(h) = σ̃α(h) otherwise. Let t be the root of à and let u = τ −1 (t). It
is easy to show (as is done in the proof of Lemma 6.8) that β(h) = β̃(h) as soon
as β̃(h) 6= t. The tree (A, (I, O)) is oriented from-root-to leaves, hence it is leftconnected. Thus, by Lemma 6.1, for any half-edge h ∈ H there exists an integer
q > 0 such that β q (h) = t. By taking the least such integer q, one gets β̃ q−1 (h) =
β q−1 (h) ∈ {u, α(u)}. Since u is in O, one gets β̃(α(u)) = β̃(u) = σ̃(u) = r. Hence,
β̃ q (h) = r. By Lemma 6.1, this implies that (M̃ , (I, O)) is left-connected.
A BIJECTION FOR COVERED MAPS
27
−1
We now study the restrictions π̃◦ ≡ π̃|I ′ and π̃• ≡ π̃|O
′ . Recall that the map B̃ is
bipartite with white vertices incident to half-edges in I ′ and black vertices incident
to half-edges in O′ . Hence, π̃ = π̃◦ π̃•−1 .
′
We first prove that the permutation π̃◦ is equal to σ̃|I
′ . We consider a half-edge
k
′
′
h in I ′ . By definition of restrictions, σ̃|I
(h) for a positive integer k such
′ (h) = σ̃
j
that for all 0 < j < k, the half-edge σ̃ ′ (h) is in O′ . Moreover, the permutations
′
′ k−1 ′
τ̃ ′ and σ̃ ′ coincide on O′ . Thus, σ̃|I
(σ̃ (h)). By (20), σ̃ ′ = τ̃ ′ π̃◦ , hence
′ (h) = τ̃
k
′
′
′
′
σ̃|I
(π̃◦ (h)). Therefore, σ̃|I
′ (h) = τ̃
′ (h) is a half-edge in I contained in the cycle of
′
′
τ̃ containing the half-edge π◦ (h) (which is in I ). Since (I, O) is the root-to-leaves
orientation of Ã, every cycle of τ̃ ′ contains exactly one half-edge in I ′ (except for
′
the cycle made of o alone). Thus, σ̃|I
′ (h) = π̃◦ (h).
We now prove that the permutation π̃• is equal to φ̃′|O′ , where φ̃′ = (i, o)σ̃ ′ α′ .
We consider the face-permutation ψ̃ = π̃α′ of B̃. Since π̃ = π̃◦ π̃•−1 one gets
−1
−1
′
′ −1
′
ψ̃ −1 = α′ π̃• π̃◦−1 , hence ψ̃|I
and finally, π̃• = α′ ψ̃|I
′ = α π̃• α π̃◦
′ π̃◦ α . We now
use the property (iv) of the coherently labelled pair (Ã, B̃). This property reads
−1
′ ′
′
′
′ ′
′
′
′
ψ̃|I
′ = α ϕ̃|O ′ α , where ϕ̃ = (i, o)τ̃ α . Thus, π̃• = ϕ̃|O ′ α π̃◦ α . We now consider a
k
half-edge h in O′ . By definition of restrictions, π̃• (h) = ϕ̃′ (α′ π̃◦ α′ (h)), where k is
the least positive integer k such that ϕ̃′ k (α′ π̃◦ α′ (h)) is in O′ . Moreover, the permutations ϕ̃′ = (i, o)τ̃ ′ α′ and φ̃′ = (i, o)σ̃ ′ α′ coincide on I ′ (since σ̃ ′ and τ̃ ′ coincide on
O′ ). Thus, π̃• (h) = φ̃′
′ k−1
that φ̃
k−1
(ϕ̃′ α′ π̃◦ α′ (h)) , where k is the least positive integer k such
(ϕ̃′ α′ π̃◦ α′ (h)) is in O′ . Moreover, ϕ̃′ α′ π̃◦ α′ (h) = (i, o)σ̃ ′ α′ (h) ≡ φ̃′ (h)
k
since ϕ̃′ = (i, o)τ̃ ′ α′ and π̃◦ = τ̃ ′−1 σ̃ ′ by (20). Thus, π̃• (h) = φ̃′ (h) = φ̃′|O′ (h).
′
′
Given that π̃◦ ≡ π̃|I ′ = σ̃|I
′ and π̃• ≡ π̃|O ′ = φ̃|O ′ , it is clear from the defini-
tion of the mapping Λ that Λ(M̃ , (I, O)) = (Ã, B̃). This concludes the proof of
Proposition 6.15.
7. Labeled and blossoming mobiles, and link with known bijections
In this section we present two alternative ways of encoding the image of the
bijection Ψ (pairs made of a tree and a mobile) and we describe the bijection Ψ
in terms of these encodings. These alternative descriptions are particularly useful
for studying the specializations of the bijection Ψ, and its relation with known
bijections. In particular we will use them to prove that Ψ can be specialized to a
classical bijection by Bouttier et al [8]. These descriptions are also used in [5, 6]
in order to define a “master bijection” which can be specialized to a bijection Ψd ,
d ≥ 3 for planar maps of girth d.
7.1. Labeled mobiles and blossoming mobiles. We first define the degreecode and height-code of a tree. Let A be a rooted plane tree with n edges. Let
v0 , v1 , . . . , vn be the vertices in counterclockwise order of appearance around the tree
(with v0 being the root). The height-code of the tree A is the sequence c0 , . . . , cn ,
where cj is the height of the vertex vj (the number of edges on the path from v0
to vj ). The degree-code (or Lukasiewicz code) is the sequence d0 , . . . , dn , where
dj is the number of children of vj . The height- and degree-code for the tree A
represented in Figure 12 are respectively 0123212 and 2210010. It is well known that
28
O. BERNARDI AND G. CHAPUY
the height-code or degree-code both determine the tree A. Moreover, a sequence of
non-negative integers c0 , . . . , cn is a height-code if and only if
c0 = 0 and ∀i < n, 0 < ci+1 ≤ ci + 1,
and a sequence of non-negative integers d0 , . . . , dn is a degree-code if and only if
n
X
i=0
(di − 1) = −1 and ∀k < n,
k
X
(di − 1) ≥ 0.
i=0
We can now give alternative descriptions of a pair (A, B) by encoding the tree A
as some decorations added to the mobile B (the decorations corresponding either
to the height- or degree-code of A). We consider the usual black and white coloring
of the mobile B (with the root-vertex being white). We say that the mobile B is
corner-labelled if a non-negative number called label is attributed to each of the
n + 1 white corners. The mobile B is well-corner-labelled if the root-corner has
label 0, all other corners have positive labels and the labels do not increase by more
than 1 from a corner to the next one in clockwise direction around B. Equivalently,
B is well-corner-labelled if the sequence of corners encountered in clockwise order
around the mobile starting from the root-corner is the height-code of a tree. A
well-corner-labelled mobile is shown in Figure 12(b). We now consider mobiles with
buds, that is, dangling half-edges. A blossoming mobile is a mobile B together with
some outgoing buds glued in each white corners. The sequence of buds encountered
in clockwise order around the mobile is the sequence d0 , d1 , . . . , dn where di is the
number of buds in the ith corner of B (in clockwise order starting from the root).
The blossoming mobile is balanced if its sequence of buds is the degree-code of
a tree. A balanced blossoming mobile is shown in Figure 12(c). Since both the
height- and degree-code (made of n + 1 integers) determine a plane tree (with n
edges) the following result is obvious.
Lemma 7.1. The three following sets are in bijection:
• pairs (A, B) made of a plane tree A and a mobile B with respectively n and
n + 1 edges,
• well-corner-labelled mobiles with n + 1 edges,
• balanced blossoming mobiles with n + 1 edges.
Lemma 7.1 is illustrated in Figure 12 (top part). By Theorem 4.4, well-cornerlabelled mobiles with n+1 edges, and balanced blossoming mobiles with n+1 edges
are in bijections with covered maps. We now describe the folding step in terms of
these objects.
Let (A, B) be a pair made of a plane tree A and a mobile B with respectively n
and n+1 edges. Recall from Section 6 the topological description of the folding step
Ω = Λ−1 for (A, B): if the vertices of the tree A are denoted v0 , v1 , . . . , vn in counterclockwise order around A and the white corners of B are denoted y0 , y1 , . . . , yn
in clockwise order around B, then the partially folded map N is obtained by gluing
the first-corner xj of the vertex vj to the corner white yj of B (see Figure 11). The
oriented map (M, (I, O)) = Ω(A, B) is then obtained from the partially folded map
N by deleting the edges and black vertices of B.
Now let (B, ℓ) be the well-corner-labelled mobile, where ℓ denotes the function
associating a label to each white corner of B. For j = 1 . . . n, we denote by yj′
be the last corner of B having label ℓ(yj ) − 1 appearing before yj in clockwise
A BIJECTION FOR COVERED MAPS
x3
x6
x4
x5
y3
y1 y
2
x2
y6
1
y4
3
2
2
2
y5
x1
x0
1
0
y0
Ω′
Ω
1
(a)
29
Ω′′
2
(b)
(c)
Figure 12. Three equivalent representations of a pair (A, B) and
descriptions of the folding step: (a) a pair consisting of a tree
and a mobile; (b) a well-corner-labelled mobile; (c) a balanced
blossoming mobile.
direction around B. Because (B, ℓ) is well-labelled, the corner yj′ always exists and
appears between the root-corner and the corner yj in clockwise order around B.
The partially folded map N ′ associated to (B, ℓ) is defined as the map obtained
from B by adding n edges sequentially: for j = 1 . . . n, a directed edge ej of N ′
is created from the corner yj′ to the corner yj (there is a unique way to draw this
edge without crossing). This procedure is represented in Figure 12(b). The (easy)
proof of the following result is omitted:
Proposition 7.2. If (B, ℓ) is the well-corner-labelled mobile corresponding with
the pair (A, B) (i.e. ℓ(y0 )ℓ(y1 ) . . . ℓ(yn ) is the height-code of A), then the partially
folded maps N and N ′ coincide.
~ be a balanced blossoming mobile. Let B
~ ′ be the fully blossoming
Let now B
~ by inserting an ingoing
mobile with ingoing and outgoing buds obtained from B
~ following an edge of B in clockwise order around the
bud in each white corner of B
~ ′ is represented in solid lines in the bottom part of Figure 12(c)).
white vertex (B
~ is balanced, the sequence of outgoing and ingoing
Because the blossoming mobile B
~ ′ (starting from the root-corner) is a parenthesis
buds in clockwise order around B
system (if outgoing and ingoing buds are seen respectively as opening and closing
parentheses). Hence, there is a unique way of pairing each outgoing bud to an
ingoing bud following it without creating any crossings. The partially folded map N ′
~ ′ by performing
associated to the blossoming mobile B̄ is the map obtained from B
these pairings. The result is represented in Figure 12(c). The (easy) proof of the
following proposition is omitted:
30
O. BERNARDI AND G. CHAPUY
~ is the blossoming mobile associated with the pair (A, B)
Proposition 7.3. If B
~ is the degree-code of A), then the partially folded
(i.e. the sequence of buds of B
maps N and N ′ coincide.
We mention that the description of the bijection Ψ in terms of blossoming mobiles
(which is used in [5, 6]) is convenient for controlling the degrees of the covered maps:
the degree of the vertices in a covered map corresponds to the degree of the white
vertices in the associated blossoming mobile.
7.2. Link with the bijection of Bouttier, Di Francesco and Guitter [8].
In [8] Bouttier, Di Francesco and Guitter defined a bijection between bipartite
maps and well-labelled mobiles 3. A well-labelled mobile is a well-corner-labelled
mobile such that the labels coincide around each white vertices, that is, any two
corners incident to the same vertex have the same label. An example is given in
Figure 13(c). Observe that well-labelled mobiles are equivalently defined as mobiles
with a label ℓ(v) associated to each vertex v satisfying:
• the root-vertex has label 0 and degree 1, while other white vertices have
positive labels,
• the increase between the labels of two consecutive white vertices in clockwise order around a black vertex is at most 1.
We now show that the bijection of Bouttier et al. can be obtained as a spe−1
cialization of the unfolding mapping Λ′ = Ω′ . We first recall some definitions.
The distance between two vertices of a map is the minimum number of edges on
paths between them. We denote by d(v) the distance of a vertex v from the rootvertex. It is easy that for bipartite maps, any pair of adjacent vertices u, v satisfies
|d(u) − d(v)| = 1. Thus one can define the geodesic orientation by requiring that
the value of d increases in the direction of every edge. The geodesic orientation is
indicated in Figure 13(b). We now state the main result of this subsection.
Proposition 7.4. The geodesic orientation of a bipartite map is left-connected.
Moreover, the unfolding mapping Λ′ induces a bijection between the set of bipartite
maps (with n edges and genus g) endowed with their geodesic orientation and the
set of well-labelled mobiles (with n + 1 edges and genus g). This bijection coincides
with the bijection described for bipartite maps in [8].
Proof. Let (M, (I, O)) be a bipartite map endowed with its geodesic orientation.
We first prove that the geodesic orientation is left-connected by using Lemma 6.1
concerning the backward function β. Clearly, for any half-edge h incident to a
non-root vertex v, there exists an integer p > 0 such that the half-edge β p (h) is
incident to a vertex u satisfying d(u) = d(v) − 1 (because there are ingoing edges
incident to v, and they all join v to a vertex u satisfying the property). Thus, there
exists q > 0 such that the half-edge β q (h) is incident to the root-vertex. Moreover,
for any half-edge h′ incident to the root-vertex there exists an integer r > 0 such
that the half-edge β r (h′ ) is the root (because the root-vertex is only incident to
outgoing half-edges). Therefore, by Lemma 6.1, the geodesic orientation is leftconnected. We now show that the corner-labelled mobile (B, ℓ) = Λ′ (M, (I, O)) is
well-labelled. Let v be a vertex of M and let v1 , . . . , vk be the vertices of the tree
A = Λ1 (M, (I, O)) resulting from unfolding the vertex v. Clearly, any directed path
3Strictly speaking, the bijection in [8] only describes the planar case. But is was explained
in [12] how to extend it to higher genera.
A BIJECTION FOR COVERED MAPS
31
from the root-vertex to v in M has length d(v). Hence, for all i ∈ {1, . . . , k} every
directed path from the root-vertex to vi in A has length d(v). Hence, the label ℓ
of every corner of the white vertex v of the mobile B is equal to d(v). Thus, the
corner-labelled mobile (B, ℓ) is a well-labelled mobile.
Conversely, let (B, ℓ) be a well-labelled mobile, let (M, (I, O)) = Ω′ (B, ℓ) be the
corresponding left-connected map. We want to prove that M is bipartite and (I, O)
is the geodesic orientation. By definition of the folding Ω′ any edge of M goes from
a white vertex u to a white vertex v satisfying ℓ(v) = ℓ(u) + 1. Hence, reasoning
on the parity of labels shows that M is bipartite. In order to prove that (I, O) is
the geodesic orientation, it suffices to prove that the label function ℓ is equal to the
distance function d. Let v be a non-root vertex. On one hand, one gets d(v) ≥ ℓ(v)
from the fact that labels cannot decrease by more than one when following an edge
of M (hence the root-vertex cannot be reached by following less than ℓ(v) edges).
On the other hand, one gets d(v) ≤ ℓ(v) from the fact that any non-root vertex of
M is adjacent to a vertex having a smaller label (by definition of the folding step
Ω′ ). Thus d = ℓ and the orientation is geodesic.
(a)
(b)
2
1
2
1
2
3
0
1
2
(c)
Figure 13. (a) The rightmost BFS tree. (b) The geodesic orientation. (c) The associated well-labelled mobile.
In the remaining of this section, we complete the picture by characterizing the
unicellular submap of a bipartite map which corresponds to the geodesic orientation
(by the orientation step ∆). A spanning tree is said BFS (for Breadth-First-Search)
if for any vertex v, the distance d(v) is equal to the height in the spanning tree.
Definition 7.5. The rightmost BFS tree is the spanning tree T obtained by the
following procedure:
Initialization: Set every vertex to be alive. Set the tree T as the tree containing
the root-vertex of M and no edge.
Core: Consider the alive vertex v which has been in the tree T for the longest
time and set it dead. Inspect the half-edges incident to v in counterclockwise order
(starting from the root if v is the root-vertex, and starting from the half-edge
following the edge of T leading v to its parent otherwise) and whenever a half-edge
leads to a vertex not in the tree T add this vertex and the edge to T .
Repeat until all vertices are dead.
End: Return the spanning tree T .
The rightmost BFS tree is indicated in Figure 13(a). We omit the (easy) proof
of the following result.
32
O. BERNARDI AND G. CHAPUY
Proposition 7.6. Let (M, S) be a bipartite covered map, and let (M, (I, O)) =
∆(M, S) be the associated left-connected map. The orientation (I, O) is geodesic if
and only if M|S is the rightmost BFS tree of M .
8. Duality.
Recall from Section 3 that the dual of a covered map is a covered map. In this section, we explore the properties of the bijection Ψ with respect to duality. Throughout this section, we consider a covered map (M, S), where the map M = (H, σ, α)
has root r and face-permutation φ = σα. We denote (M, (I, O)) = ∆(M, S) and
(A, B) = Ψ(M, S).
Lemma 8.1 (Duality at the orientation step). The oriented map associated to the
dual covered map is the dual oriented map, that is to say, ∆(M ∗ , S̄) = (M ∗ , (O, I)).
Lemma 8.1 is illustrated in Figure 14.
Proof. Recall that the the submaps M|S and M|∗S̄ have the same motion functions,
hence define the same appearance order on H. Thus, Lemma 8.1 immediately
follows from the definition of the mapping ∆.
a4
b̄3
b3
a3 b̄2
ā3
ā1
b̄3
ā4
a4
a1
ā3
b2
b̄2
a2
ā4
b3 a 3
ā2
b2
ā2
b̄1
b1
(a)
ā1
a1
a2
b̄1
b1
(b)
(c)
Figure 14. (a) The oriented map (M, (I, O)) = ∆(M, S) associated to the covered map represented in Figure 4(a). (b) Topological
construction of the dual: each oriented edge of M is crossed by the
the dual oriented edge of M ∗ from left to right. (c) The oriented
map (M ∗ , (O, I)).
We now explore the properties of the unfolding step with respect to duality.
We denote A = (H, τ, α) = Ψ1 (M, S) and B = (H ′ , π, α′ ) = Ψ2 (M, S), where
H ′ stands for H ∪ {i, o} and i is the root of the mobile B. We also denote by
A⋆ = (H, τ ⋆ , α) = Ψ1 (M ∗ , S̄) and B ⋆ = (H ′ , π ⋆ , α′ ) = Ψ2 (M ∗ , S̄), the plane tree
and mobile associated to the dual covered map (M ∗ , S̄). We shall prove the existence of two independent mappings Υ and Ξ such that A⋆ = Υ(A) and B ⋆ = Ξ(B).
In words, the duality acts component-wise on the plane tree and the mobile.
Proposition 8.2 (Duality and the mobile). Let (M, S) be a covered map and let
(M ∗ , S̄) be the dual covered map. If the mobile B = Ψ2 (M, S) is denoted (H ′ , π, α′ )
and has root i, then the mobile B ⋆ = Ψ2 (M ∗ , S̄) is the map (H ′ , π −1 , α′ ) with root
o = α(i).
A BIJECTION FOR COVERED MAPS
33
Figure 15. Simultaneous unfolding of the oriented map of Figure 14 and of its dual.
Proposition 8.2 is illustrated in Figure 16. It implies that the mobile B ⋆ is
entirely determined by the mobile B.
Proof. The map M has vertex-permutation σ and face-permutation φ, while the
map M ∗ has vertex-permutation σ ⋆ = φ and face-permutation φ⋆ = σ. We denote
∆(M, S) = (M, (I, O)), so that B = Λ(M, (I, O)) and B = Λ(M ∗ , (I ⋆ , O⋆ )), where
I ⋆ = O and O⋆ = I by Lemma 8.1. We adopt the notations i, o, I ′ , O′ , σ ′ , φ′ , π◦ ,
⋆
π• , π of Section 4 for defining B and adopt the corresponding notations i⋆ , o⋆ , I ′ ,
′⋆
′⋆
′⋆
⋆
⋆
⋆
⋆
⋆
⋆
O , σ , φ , π◦ , π• , π for defining B . We choose i = o and o = i, so that
⋆
⋆
⋆
⋆
I ′ ≡ I ⋆ ∪ {i⋆ } = O′ , O′ ≡ O⋆ ∪ {o⋆ } = I ′ , σ ′ = φ′ and φ′ = σ ′ . From this,
⋆
′⋆ ′⋆
′
⋆
′⋆
′ ′
it follows that π◦ ≡ σ |I = φ |O′ = π• and π• ≡ φ |O′ ⋆ = σ |I = π◦ and finally
π ⋆ ≡ π•⋆ π◦⋆ −1 = π◦ π•−1 = π −1 . Lastly, the root of B ⋆ is i⋆ = o.
Ξ
Figure 16. The mapping Ξ between the mobile B = Ψ2 (M, S)
associated to the covered map (M, S) of Figure 14 and the mobile
B ⋆ = Ψ2 (M ∗ , S̄) associated to the dual covered map.
We now explicit the relation between the trees A and A⋆ in terms of their codes.
Proposition 8.3 (Duality and the tree). If the height-code of A = Ψ1 (M, S) is
c0 , . . . , cn , then the degree-code of A⋆ = Ψ1 (M ∗ , S̄) is d0 , . . . , dn , where dn−j =
cj + 1 − cj+1 for j = 1, . . . , n−1 and d0 = cn .
Recall that a tree is completely determined by its height-code or by its degreecode. Hence, Proposition 8.3 shows that the tree A⋆ is entirely determined by the
tree A. Observe that the mapping A 7→ A⋆ is an involution since duality of covered map is an involution. A topological version of this mapping is illustrated in
Figure 17(b), where the two trees A and A⋆ are represented simultaneously in the
way they interlace around the mobile’s face. The rest of this section is devoted to
34
O. BERNARDI AND G. CHAPUY
the proof of Proposition 8.3.
b1
b̄2
b̄1
ā2
b3
b̄3
a2
ā3
a1
b2
a3
ā4
a4
ā1
b̄2
ā1
a1
(a)
(b)
a2
b2 ā2
b1
a3
ā3
a4
ā4
b̄1
b̄3
b3
(c)
Figure 17. (a) The plane tree A = Ψ1 (M, S) associated to the
covered map of Figure 14. (b) Topological construction of the tree
A⋆ = Ψ1 (M ∗ , S̄): in the mobile face, the trees A and A⋆ interlace
in such a way that each edge of A is crossed by exactly one edge
of A⋆ . (c) The plane tree A⋆ .
We denote by t the root of the tree A = (H, τ, α) and by t⋆ the root of the tree
A = (H, τ ⋆ , α). We also adopt the notations σ ′ , φ′ , π◦ , π• , π, τ ′ of Section 4
⋆
⋆
for the tree A and adopt the corresponding notations σ ′ , φ′ , π◦ ⋆ , π• ⋆ , π ⋆ , τ ′⋆
⋆
′
′ ′
for the tree A . Lastly, we denote ϕ = τ α, ϕ = (i, o)τ α , ϕ⋆ = τ ⋆ α and
ϕ′⋆ = (i, o)τ ′⋆ α′ . Recall that Lemma 6.11 describes the (simple) link existing
between the permutations τ and τ ′ and between ϕ and ϕ′ .
⋆
⋆
⋆
Lemma 8.4. The permutations ϕ′ and ϕ′ are related by ϕ′ |O′ = (α′ ϕ′ α′ )−1
|O′ .
For the example in Figure 17, one gets ϕ′ |O′ = (o, a1 , b̄2 , a2 , b̄1 , a3 , a4 , b̄3 ) and
= (b3 , ā4 , ā3 , b1 , ā2 , b2 , ā1 , i).
⋆
ϕ′ |I ′
Proof. By Proposition 6.12, the face-permutation ψ = πα of the mobile B satis⋆
−1 ′
⋆
fies ϕ′ |O′ = α′ ψ|I
gives ϕ′ |I ′ =
′ α . The same property applied to the mobile B
′
⋆
⋆ ′
α′ ψ ⋆ −1
|O′ α , where ψ = π α is the face-permutation of B. This gives
⋆
⋆
′
′ −1 ′
⋆
(α′ ϕ′ α′ )−1
|O′ = α (ϕ )|I ′ α = ψ |O′ .
Moreover, by Proposition 8.2, π ⋆ = π −1 , so that ψ ⋆ = π −1 α′ = α′ ψ −1 α. Hence,
⋆
−1 ′
⋆
′ −1
′
(α′ ϕ′ α′ )−1
α)|O′ = α′ ψ|I
′ α = ϕ |O ′ .
|O′ = ψ |O′ = (α ψ
′
Lemma 8.5. The permutations ϕ and τ
′⋆
are related by τ
′⋆
′
=ϕ
−1
ϕ′ |O′ .
⋆
′
′ ′
Proof. By definition, τ ′ = σ ′⋆ π◦⋆ −1 = φ′ φ′−1
|O′ , where φ = (i, o)σ α . We want
−1
−1
−1
to prove φ′ φ′ |O′ = ϕ′ ϕ′ |O′ , or equivalently, φ′|O′ φ′−1 = ϕ′|O′ ϕ′
(by taking the
′
′ ′
′
inverse). Observe that the permutations φ = (i, o)σ α and ϕ = (i, o)τ ′ α′ coincide
on I ′ (since σ ′ and τ ′ coincide on O′ ). We now consider a half-edge h in H ′ . Suppose
−1
−1
first that φ′−1 (h) is in I ′ . In this case, φ′ (h) = ϕ′ (h) (since φ′ and ϕ′ coincide
A BIJECTION FOR COVERED MAPS
on I ′ ), hence φ′|O′ φ′−1 (h) = φ′
′ −1
−1
(h) = ϕ′
−1
(h) = ϕ′|O′ ϕ′
′ −1
′
−1
(h). Suppose now that
φ (h) is in O . Observe that ϕ (h) is also in O (since φ and ϕ′ coincide on I ′ ).
k
Moreover, by definition of restrictions, φ′|O′ φ′−1 (h) = φ′ (h), where k ≥ 0 is such
k
′
35
′
j
that φ′ (h) ∈ O′ and φ′ (h) ∈ I ′ for all 0 ≤ j < k. Since φ′ and ϕ′ coincide on I ′ ,
we get φ′ j (h) = ϕ′ j (h) for 0 ≤ j ≤ k. Thus, φ′|O′ φ′−1 (h) = ϕ′ k (h) and where k ≥ 0
k
j
is such that ϕ′ (h) ∈ O′ and ϕ′ (h) ∈ I ′ for all 0 ≤ j < k. Hence, by definition of
−1
restrictions, φ′|O′ φ′−1 (h) = ϕ′|O′ ϕ′ (h).
Proof of Proposition 8.3. We denote by o0 = o, o1 , . . . , on the half-edges in O′ in
such a way that ϕ′ |O′ = (o, o1 , . . . , on ) and we denote ij = α(oj ) for j = 0 . . . n.
We denote by v0 , v1 , . . . , vn the vertices of A in counterclockwise order around A.
By definition (and because (I, O) is the root-to-leaves orientation of A), this means
that vj is incident to the ingoing half-edge ij for j = 1 . . . n. Therefore, the heightcode of A is c0 c1 · · · cn , where c0 = 0 and for j = 0 . . . n−1, cj+1 = cj + 1 − δj where
δj is the number of half-edges in I between the half-edge oj and oj+1 in the facepermutation ϕ (hence, also in the permutation ϕ′ ). Equivalently, for j = 0 . . . n−1,
δj ≡ cj + 1 − cj+1 is the number of half-edges in I ′ in the cycle of the permutation
−1
ϕ′ ϕ′ |O′ containing oj . We also denote δn = cn and observe that this is the number
−1
of half-edges in I ′ in the cycle of the permutation ϕ′ ϕ′ |O′ containing on .
⋆
We now consider degree-code d0 d1 · · · dn of A and want to prove that δj = dn−j
for j = 0 . . . n. Let v0⋆ , v1⋆ , . . . , vn⋆ be the vertices of A⋆ in counterclockwise order
around A⋆ . By Lemma 8.1, the root-to-leaves orientation of A⋆ is (O, I) and by
⋆
⋆
Lemma 8.4, ϕ′ |I ′ = (i, in , . . . , i2 , i1 ). Therefore, for j = 1 . . . n−1 the vertex vn−j
of
⋆
A is incident to the half-edge oj . Thus, for j = 0 . . . n−1, the number of children
⋆
dn−j of vn−j
is the number of half-edges in I ′ in the cycle of the vertex-permutation
⋆
τ containing oj (hence, also in the permutation τ ′⋆ ). For j = n also, we observe
that dn−j is the number of half-edges in I ′ in the cycle of the permutation τ ′⋆
−1
containing oj . By Lemma 8.5, τ ′⋆ = ϕ′ ϕ′ |O′ , hence δj = dn−j for j = 0 . . . n. This
concludes the proof of Proposition 8.3.
Acknowledgments. We thank Éric Fusy, Jean-François Marckert, Grégory Miermont, and Gilles Schaeffer for very stimulating discussions.
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